Final_Project_Part i+ Part ii
2023-12-10
R Markdown
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## accumulate, whendf <- readr::read_csv("paint_project_train_data.csv", col_names = TRUE)## Rows: 835 Columns: 8
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (2): Lightness, Saturation
## dbl (6): R, G, B, Hue, response, outcome
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.df %>% glimpse()## Rows: 835
## Columns: 8
## $ R <dbl> 172, 26, 172, 28, 170, 175, 90, 194, 171, 122, 0, 88, 144, …
## $ G <dbl> 58, 88, 94, 87, 66, 89, 78, 106, 68, 151, 121, 140, 82, 163…
## $ B <dbl> 62, 151, 58, 152, 58, 65, 136, 53, 107, 59, 88, 58, 132, 50…
## $ Lightness <chr> "dark", "dark", "dark", "dark", "dark", "dark", "dark", "da…
## $ Saturation <chr> "bright", "bright", "bright", "bright", "bright", "bright",…
## $ Hue <dbl> 4, 31, 8, 32, 5, 6, 34, 10, 1, 21, 24, 22, 36, 16, 26, 12, …
## $ response <dbl> 12, 10, 16, 10, 11, 16, 10, 19, 14, 25, 14, 19, 14, 38, 15,…
## $ outcome <dbl> 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1,…Part i: Exploration
#1. Visualize the distributions of variables in the data set.
#1.1 Counts for categorical variables.
ans: Here we calculate the lightness variables and saturation separately, and we also calculate them into each pair with one lightness and saturation.
counts_lightness <- df %>%
count(Lightness)
counts_saturation <- df %>%
count(Saturation)
counts_df <- df %>%
count(Lightness, Saturation)
print(counts_lightness)## # A tibble: 7 × 2
## Lightness n
## <chr> <int>
## 1 dark 117
## 2 deep 119
## 3 light 120
## 4 midtone 119
## 5 pale 121
## 6 saturated 119
## 7 soft 120print(counts_saturation)## # A tibble: 7 × 2
## Saturation n
## <chr> <int>
## 1 bright 126
## 2 gray 83
## 3 muted 126
## 4 neutral 122
## 5 pure 126
## 6 shaded 126
## 7 subdued 126print(counts_df)## # A tibble: 49 × 3
## Lightness Saturation n
## <chr> <chr> <int>
## 1 dark bright 18
## 2 dark gray 10
## 3 dark muted 18
## 4 dark neutral 17
## 5 dark pure 18
## 6 dark shaded 18
## 7 dark subdued 18
## 8 deep bright 18
## 9 deep gray 12
## 10 deep muted 18
## # ℹ 39 more rows#1.2 Histograms or Density plots for continuous variables. Are the distributions Gaussian like?
Acoording to the graphs below, they are not Gaussian like.
ggplot(df, aes(x = response)) +
geom_density(fill = "lightblue", color = "darkblue", binwidth = 10) +
labs(title = "Density Plot of Response", x = "Response")## Warning in geom_density(fill = "lightblue", color = "darkblue", binwidth = 10):
## Ignoring unknown parameters: `binwidth`
ggplot(df, aes(x = R)) +
geom_density(fill = "lightblue", color = "darkblue", binwidth = 10) +
labs(title = "Density Plot of R", x = "R")## Warning in geom_density(fill = "lightblue", color = "darkblue", binwidth = 10):
## Ignoring unknown parameters: `binwidth`
ggplot(df, aes(x = G)) +
geom_density(fill = "lightblue", color = "darkblue", binwidth = 10) +
labs(title = "Density Plot of G", x = "G")## Warning in geom_density(fill = "lightblue", color = "darkblue", binwidth = 10):
## Ignoring unknown parameters: `binwidth`
ggplot(df, aes(x = B)) +
geom_density(fill = "lightblue", color = "darkblue", binwidth = 10) +
labs(title = "Density Plot of B", x = "B")## Warning in geom_density(fill = "lightblue", color = "darkblue", binwidth = 10):
## Ignoring unknown parameters: `binwidth`
ggplot(df, aes(x = Hue)) +
geom_density(fill = "lightblue", color = "darkblue", binwidth = 10) +
labs(title = "Density Plot of Hue", x = "Hue")## Warning in geom_density(fill = "lightblue", color = "darkblue", binwidth = 10):
## Ignoring unknown parameters: `binwidth`
#2.Condition (group) the continuous variables based on the categorical variables. #2.1 Are there differences in continuous variable distributions and continuous variable summary statistics based on categorical variable values?
library(reshape2)##
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## smiths# 使用 melt 函數將 R, G, B 轉換為一個變數
df_melted <- melt(df, id.vars = c("Lightness"), measure.vars = c("R", "G", "B"))
# 繪製盒狀圖
ggplot(df_melted, aes(x = Lightness, y = value, fill = variable)) +
geom_boxplot() +
labs(title = "Boxplot of R, G, B Grouped by Lightness") +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
# 使用ANOVA進行分組比較
model_anova <- aov(R ~ Lightness, data = df)
anova_result <- summary(model_anova)
print(anova_result)## Df Sum Sq Mean Sq F value Pr(>F)
## Lightness 6 1460742 243457 157.3 <2e-16 ***
## Residuals 828 1281177 1547
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1In your results, the p-value Pr(>F) is less than 0.05 (common significance level), so we reject the null hypothesis. This implies that there is a statistically significant difference between different levels of Lightness.
#2.2 Are there differences in continuous variable distributions and continuous variable summary statistics based on the binary outcome？
summary(df$response)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 6.0 26.0 51.0 48.6 72.0 87.0# Summary statistics for 'response' based on 'outcome'
summary(df$response[df$outcome == 1]) # Replace 1 with the actual code for the positive outcome## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 6.00 17.00 46.00 45.22 70.00 85.00summary(df$response[df$outcome == 0]) # Replace 0 with the actual code for the negative outcome## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 6.00 27.00 52.50 49.61 73.00 87.00the first set of answers focuses on summarizing the overall distribution of a continuous variable, while the second set of answers explores how the distribution of the continuous variable differs based on a binary outcome variable. The second set of answers involves comparing the distribution of the continuous variable between different groups defined by the binary outcome.
#3.Visualize the relationships between the continuous inputs, are they correlated?
mod01 <- lm( response ~ R*G*B*Hue, data = df )mod01%>% coefplot::coefplot()+
theme(legend.position = 'none')
As we see below all features are not statistically significant except Hue, because Hue doesn’t contain zero.
# 使用散點圖視覺化連續輸入變量之間的關係
pairs(df[, c("R", "G", "B", "Hue")], pch = 16, col = "blue", main = "Scatterplot Matrix")
# 計算相關性矩陣
cor_matrix <- cor(df[, c("R", "G", "B", "Hue")])
# 視覺化相關性矩陣
heatmap(cor_matrix, annot = TRUE, cmap = "Blues", main = "Correlation Matrix")## Warning in plot.window(...): "annot" is not a graphical parameter## Warning in plot.window(...): "cmap" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "annot" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "cmap" is not a graphical parameter## Warning in title(...): "annot" is not a graphical parameter## Warning in title(...): "cmap" is not a graphical parameter
4.Visualize the relationships between the continuous outputs (response and the LOGIT-transformed response, y) with respect to the continuous INPUTS. 4.1Can you identify any clear trends? Do the trends depend on the categorical INPUTS?
dfii <- df %>%
mutate(y = boot::logit( (response - 0) / (100 - 0) ) ) %>%
select(R, G, B,
Lightness, Saturation, Hue,
y)
dfii %>% glimpse()## Rows: 835
## Columns: 7
## $ R <dbl> 172, 26, 172, 28, 170, 175, 90, 194, 171, 122, 0, 88, 144, …
## $ G <dbl> 58, 88, 94, 87, 66, 89, 78, 106, 68, 151, 121, 140, 82, 163…
## $ B <dbl> 62, 151, 58, 152, 58, 65, 136, 53, 107, 59, 88, 58, 132, 50…
## $ Lightness <chr> "dark", "dark", "dark", "dark", "dark", "dark", "dark", "da…
## $ Saturation <chr> "bright", "bright", "bright", "bright", "bright", "bright",…
## $ Hue <dbl> 4, 31, 8, 32, 5, 6, 34, 10, 1, 21, 24, 22, 36, 16, 26, 12, …
## $ y <dbl> -1.9924302, -2.1972246, -1.6582281, -2.1972246, -2.0907411,…mod02 <- lm( y ~ R*G*B*Hue, data = dfii )
mod02 %>% coefplot::coefplot()+
theme(legend.position = 'none')
5.How can you visualize the behavior of the binary outcome with respect to the continuous inputs? How can you visualize the behavior of the binary outcome with respect to the categorical INPUTS?
library(reshape2)
# 使用 melt 函數將 R, G, B 轉換為一個變數
df_melted <- melt(df, id.vars = c("outcome"), measure.vars = c("R", "G", "B"))
# 繪製盒狀圖
ggplot(df_melted, aes(x = outcome, y = value, fill = variable)) +
geom_boxplot() +
labs(title = "Boxplot of R, G, B Grouped by outcome") +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
# Create a boxplot for the binary outcome with respect to Lightness and Saturation
ggplot(df, aes(x = Lightness, y = response, fill = factor(outcome))) +
geom_boxplot() +
labs(title = "Boxplot of Response with Respect to Lightness and Outcome", x = "Lightness", y = "Response", fill = "Outcome") +
facet_wrap(~ Saturation, scales = "free", ncol = 2) # Facet by Saturation
Part ii: Regression - iiA) Linear models
Before using more advanced methods, you need to develop a baseline understanding for the behavior of the LOGIT-transformed response as a function of the inputs using linear modeling techniques.
df_standard <- dfii
# Standardization function
standardize <- function(x) {
return ((x - mean(x)) / sd(x))
}
# Apply the function to the variables
df_standard$R <- standardize(dfii$R)
df_standard$G <- standardize(dfii$G)
df_standard$B <- standardize(dfii$B)
df_standard$Hue <- standardize(dfii$Hue)
df_standard$y <- standardize(dfii$y)
df_standard %>% glimpse()## Rows: 835
## Columns: 7
## $ R <dbl> -0.19790120, -2.74419521, -0.19790120, -2.70931447, -0.2327…
## $ G <dbl> -2.3619736, -1.7631189, -1.6433480, -1.7830807, -2.2022790,…
## $ B <dbl> -1.7994266, -0.1706872, -1.8726283, -0.1523868, -1.8726283,…
## $ Lightness <chr> "dark", "dark", "dark", "dark", "dark", "dark", "dark", "da…
## $ Saturation <chr> "bright", "bright", "bright", "bright", "bright", "bright",…
## $ Hue <dbl> -1.3548215, 1.3198239, -0.9585777, 1.4188848, -1.2557605, -…
## $ y <dbl> -1.5899718, -1.7628901, -1.3077880, -1.7628901, -1.6729807,…Use lm() to fit linear models. You must use the following: #A1. Intercept-only model – no INPUTS!
data <- data.frame(logit_response = dfii$y)
intercept_only_model <- lm(logit_response ~ 1, data = data)
summary(intercept_only_model)##
## Call:
## lm(formula = logit_response ~ 1, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.6422 -0.9366 0.1494 1.0538 2.0103
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.10936 0.04099 -2.668 0.00777 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.184 on 834 degrees of freedomfit_lm_01 <- lm(y ~ 1, data =df_standard)
fit_lm_01 %>% summary()##
## Call:
## lm(formula = y ~ 1, data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.2309 -0.7908 0.1261 0.8898 1.6974
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.305e-17 3.461e-02 0 1
##
## Residual standard error: 1 on 834 degrees of freedom#A2. Categorical variables only – linear additive
fit_lm_02 <- lm(y ~ Lightness + Saturation, data = df_standard)
fit_lm_02 %>% summary()##
## Call:
## lm(formula = y ~ Lightness + Saturation, data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.90968 -0.22814 -0.01335 0.19000 1.35945
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.30251 0.04224 -30.837 < 2e-16 ***
## Lightnessdeep 0.46151 0.04455 10.360 < 2e-16 ***
## Lightnesslight 2.35306 0.04446 52.931 < 2e-16 ***
## Lightnessmidtone 1.51244 0.04455 33.952 < 2e-16 ***
## Lightnesspale 2.66009 0.04437 59.954 < 2e-16 ***
## Lightnesssaturated 1.01152 0.04455 22.707 < 2e-16 ***
## Lightnesssoft 1.98898 0.04446 44.741 < 2e-16 ***
## Saturationgray -0.47209 0.04837 -9.759 < 2e-16 ***
## Saturationmuted -0.15178 0.04310 -3.521 0.000453 ***
## Saturationneutral -0.27411 0.04346 -6.308 4.62e-10 ***
## Saturationpure 0.34271 0.04310 7.951 6.12e-15 ***
## Saturationshaded -0.25504 0.04310 -5.917 4.81e-09 ***
## Saturationsubdued -0.23723 0.04310 -5.504 4.97e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3421 on 822 degrees of freedom
## Multiple R-squared: 0.8846, Adjusted R-squared: 0.8829
## F-statistic: 525.3 on 12 and 822 DF, p-value: < 2.2e-16library(coefplot)coefplot(fit_lm_02)
#A3. Continuous variables only – linear additive
fit_lm_03 <- lm(y ~ R + G + B + Hue, data = df_standard)
coefplot(fit_lm_03)
#A4.All categorical and continuous variables – linear additive
fit_lm_04 <- lm(y ~ ., data = df_standard)
coefplot(fit_lm_04)
#A5. Interaction of the categorical inputs with all continuous inputs main effects
fit_lm_05 <- lm(y ~ (Lightness + Saturation) * (R + G + B + Hue), data = df_standard)
coefplot(fit_lm_05)
#A6. Add categorical inputs to all main effect and all pairwise interactions of continuous inputs
fit_lm_06 <- lm(y ~ Lightness + Saturation + (R + G + B + Hue)^2, data = df_standard)
fit_lm_06 %>% summary()##
## Call:
## lm(formula = y ~ Lightness + Saturation + (R + G + B + Hue)^2,
## data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.236008 -0.038741 -0.003913 0.035839 0.241332
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.030689 0.018823 -1.630 0.103407
## Lightnessdeep 0.043867 0.010872 4.035 5.98e-05 ***
## Lightnesslight -0.067735 0.025544 -2.652 0.008166 **
## Lightnessmidtone -0.085469 0.019500 -4.383 1.32e-05 ***
## Lightnesspale 0.036528 0.027182 1.344 0.179377
## Lightnesssaturated -0.007675 0.015221 -0.504 0.614230
## Lightnesssoft -0.099784 0.022806 -4.375 1.37e-05 ***
## Saturationgray -0.035763 0.011384 -3.142 0.001741 **
## Saturationmuted -0.017965 0.008552 -2.101 0.035971 *
## Saturationneutral -0.044623 0.009441 -4.727 2.69e-06 ***
## Saturationpure 0.025348 0.008919 2.842 0.004597 **
## Saturationshaded -0.034739 0.009117 -3.810 0.000149 ***
## Saturationsubdued -0.038154 0.008793 -4.339 1.61e-05 ***
## R 0.232483 0.006785 34.264 < 2e-16 ***
## G 0.749420 0.009083 82.512 < 2e-16 ***
## B 0.135239 0.007819 17.296 < 2e-16 ***
## Hue -0.004240 0.005137 -0.825 0.409434
## R:G 0.028167 0.004905 5.742 1.32e-08 ***
## R:B -0.010631 0.007314 -1.453 0.146487
## R:Hue -0.041204 0.007204 -5.720 1.50e-08 ***
## G:B 0.053081 0.006202 8.559 < 2e-16 ***
## G:Hue 0.027295 0.008677 3.146 0.001717 **
## B:Hue 0.007130 0.010809 0.660 0.509713
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.06666 on 812 degrees of freedom
## Multiple R-squared: 0.9957, Adjusted R-squared: 0.9956
## F-statistic: 8494 on 22 and 812 DF, p-value: < 2.2e-16coefplot(fit_lm_06)
#A7. Interaction of the categorical inputs with all main effect and all pairwise interactions of continuous inputs
fit_lm_07 <- lm(y ~ (Lightness + Saturation) * (R + G + B + Hue)^2, data = df_standard)
coefplot(fit_lm_07)
#A8. 3 models with basis functions of your choice Try non-linear basis functions based on your EDA.
fit_lm_08 <- lm(y ~ (Lightness + Saturation) * (( R + G + B + Hue)^2 + I(R^2) + I(G^2) + I(B^2) + I(Hue^2)), data = df_standard)
fit_lm_08 %>% summary()##
## Call:
## lm(formula = y ~ (Lightness + Saturation) * ((R + G + B + Hue)^2 +
## I(R^2) + I(G^2) + I(B^2) + I(Hue^2)), data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.125500 -0.015352 -0.000605 0.015508 0.133380
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.740e-01 1.229e-01 -4.669 3.68e-06 ***
## Lightnessdeep 4.347e-01 1.180e-01 3.683 0.000250 ***
## Lightnesslight 5.684e-01 1.325e-01 4.289 2.07e-05 ***
## Lightnessmidtone 4.396e-01 1.242e-01 3.539 0.000430 ***
## Lightnesspale 1.024e+00 1.769e-01 5.790 1.10e-08 ***
## Lightnesssaturated 4.619e-01 1.207e-01 3.827 0.000142 ***
## Lightnesssoft 4.665e-01 1.248e-01 3.738 0.000202 ***
## Saturationgray -1.993e-02 3.240e-02 -0.615 0.538765
## Saturationmuted -1.097e-02 3.098e-02 -0.354 0.723496
## Saturationneutral -2.988e-03 3.065e-02 -0.098 0.922355
## Saturationpure -5.557e-02 4.475e-02 -1.242 0.214840
## Saturationshaded -6.139e-04 3.087e-02 -0.020 0.984140
## Saturationsubdued -1.784e-02 3.093e-02 -0.577 0.564358
## R 1.631e-01 4.856e-02 3.359 0.000829 ***
## G 4.170e-01 5.547e-02 7.517 1.90e-13 ***
## B -4.034e-01 1.302e-01 -3.097 0.002038 **
## Hue 1.331e-01 6.460e-02 2.060 0.039763 *
## I(R^2) 4.656e-02 9.126e-03 5.102 4.44e-07 ***
## I(G^2) 2.743e-02 2.873e-02 0.955 0.340111
## I(B^2) -1.243e-01 3.887e-02 -3.197 0.001456 **
## I(Hue^2) 2.758e-03 1.805e-02 0.153 0.878584
## R:G -8.968e-02 3.068e-02 -2.923 0.003584 **
## R:B -2.549e-02 2.682e-02 -0.950 0.342234
## R:Hue 2.807e-02 3.164e-02 0.887 0.375308
## G:B -1.531e-01 4.267e-02 -3.589 0.000358 ***
## G:Hue 3.767e-02 3.407e-02 1.105 0.269365
## B:Hue 8.085e-02 3.987e-02 2.028 0.043005 *
## Lightnessdeep:R -7.683e-03 4.321e-02 -0.178 0.858925
## Lightnesslight:R 1.451e-01 1.371e-01 1.058 0.290250
## Lightnessmidtone:R 5.111e-02 4.755e-02 1.075 0.282878
## Lightnesspale:R -3.972e-01 3.646e-01 -1.089 0.276491
## Lightnesssaturated:R 6.853e-02 4.376e-02 1.566 0.117829
## Lightnesssoft:R 5.585e-02 6.214e-02 0.899 0.369108
## Lightnessdeep:G 2.248e-01 5.025e-02 4.473 9.12e-06 ***
## Lightnesslight:G -2.549e-01 1.660e-01 -1.536 0.125121
## Lightnessmidtone:G 2.251e-01 5.900e-02 3.815 0.000150 ***
## Lightnesspale:G -8.384e-01 4.563e-01 -1.837 0.066623 .
## Lightnesssaturated:G 2.284e-01 5.160e-02 4.427 1.13e-05 ***
## Lightnesssoft:G 7.211e-02 8.030e-02 0.898 0.369491
## Lightnessdeep:B 5.033e-01 1.260e-01 3.996 7.18e-05 ***
## Lightnesslight:B 4.837e-01 1.612e-01 3.001 0.002799 **
## Lightnessmidtone:B 5.347e-01 1.315e-01 4.067 5.36e-05 ***
## Lightnesspale:B 4.947e-01 2.378e-01 2.080 0.037898 *
## Lightnesssaturated:B 5.630e-01 1.260e-01 4.468 9.34e-06 ***
## Lightnesssoft:B 5.238e-01 1.354e-01 3.870 0.000120 ***
## Lightnessdeep:Hue -2.048e-01 6.599e-02 -3.104 0.001994 **
## Lightnesslight:Hue -1.332e-01 7.253e-02 -1.837 0.066663 .
## Lightnessmidtone:Hue -1.589e-01 6.363e-02 -2.497 0.012772 *
## Lightnesspale:Hue -1.625e-01 8.888e-02 -1.828 0.068054 .
## Lightnesssaturated:Hue -1.444e-01 6.403e-02 -2.255 0.024445 *
## Lightnesssoft:Hue -1.594e-01 6.563e-02 -2.429 0.015411 *
## Lightnessdeep:I(R^2) -9.327e-03 8.874e-03 -1.051 0.293647
## Lightnesslight:I(R^2) -7.219e-02 7.377e-02 -0.979 0.328193
## Lightnessmidtone:I(R^2) 4.760e-03 1.779e-02 0.268 0.789132
## Lightnesspale:I(R^2) 2.988e-01 2.231e-01 1.339 0.180897
## Lightnesssaturated:I(R^2) 1.755e-03 9.818e-03 0.179 0.858179
## Lightnesssoft:I(R^2) 1.044e-02 3.702e-02 0.282 0.778111
## Lightnessdeep:I(G^2) 6.690e-02 2.053e-02 3.258 0.001180 **
## Lightnesslight:I(G^2) 3.432e-01 1.293e-01 2.654 0.008158 **
## Lightnessmidtone:I(G^2) 9.108e-02 4.472e-02 2.037 0.042087 *
## Lightnesspale:I(G^2) 2.753e-01 3.361e-01 0.819 0.413091
## Lightnesssaturated:I(G^2) 6.074e-02 2.885e-02 2.105 0.035667 *
## Lightnesssoft:I(G^2) 2.548e-01 7.069e-02 3.605 0.000337 ***
## Lightnessdeep:I(B^2) 1.410e-01 3.725e-02 3.785 0.000168 ***
## Lightnesslight:I(B^2) 2.097e-01 6.587e-02 3.184 0.001523 **
## Lightnessmidtone:I(B^2) 1.675e-01 3.893e-02 4.304 1.94e-05 ***
## Lightnesspale:I(B^2) 2.031e-01 1.088e-01 1.867 0.062397 .
## Lightnesssaturated:I(B^2) 1.667e-01 3.840e-02 4.341 1.65e-05 ***
## Lightnesssoft:I(B^2) 2.116e-01 4.521e-02 4.681 3.49e-06 ***
## Lightnessdeep:I(Hue^2) -3.804e-03 1.196e-02 -0.318 0.750587
## Lightnesslight:I(Hue^2) -4.958e-03 1.680e-02 -0.295 0.768033
## Lightnessmidtone:I(Hue^2) -6.729e-03 1.573e-02 -0.428 0.668964
## Lightnesspale:I(Hue^2) -4.647e-03 1.597e-02 -0.291 0.771202
## Lightnesssaturated:I(Hue^2) -1.369e-02 1.332e-02 -1.028 0.304457
## Lightnesssoft:I(Hue^2) -1.799e-02 1.718e-02 -1.047 0.295356
## Saturationgray:R -7.726e-03 1.336e-01 -0.058 0.953893
## Saturationmuted:R 1.801e-03 2.805e-02 0.064 0.948832
## Saturationneutral:R 6.425e-02 5.264e-02 1.221 0.222646
## Saturationpure:R 3.889e-02 3.024e-02 1.286 0.198924
## Saturationshaded:R -8.517e-03 3.989e-02 -0.214 0.830987
## Saturationsubdued:R 2.550e-02 3.297e-02 0.773 0.439627
## Saturationgray:G -1.178e-01 1.631e-01 -0.723 0.470079
## Saturationmuted:G 6.606e-04 4.410e-02 0.015 0.988053
## Saturationneutral:G -6.066e-02 8.990e-02 -0.675 0.500036
## Saturationpure:G 4.224e-02 3.922e-02 1.077 0.281977
## Saturationshaded:G 3.322e-02 6.555e-02 0.507 0.612488
## Saturationsubdued:G -4.625e-04 5.330e-02 -0.009 0.993079
## Saturationgray:B 1.423e-01 1.037e-01 1.372 0.170585
## Saturationmuted:B 2.640e-02 3.160e-02 0.835 0.403833
## Saturationneutral:B 1.006e-02 5.894e-02 0.171 0.864533
## Saturationpure:B -3.860e-02 2.836e-02 -1.361 0.174060
## Saturationshaded:B -2.382e-02 4.371e-02 -0.545 0.585942
## Saturationsubdued:B 7.099e-04 3.715e-02 0.019 0.984760
## Saturationgray:Hue 3.804e-05 3.719e-02 0.001 0.999184
## Saturationmuted:Hue 2.625e-04 2.964e-02 0.009 0.992935
## Saturationneutral:Hue 4.235e-02 3.060e-02 1.384 0.166833
## Saturationpure:Hue 3.863e-02 3.700e-02 1.044 0.296781
## Saturationshaded:Hue 2.487e-02 3.019e-02 0.824 0.410388
## Saturationsubdued:Hue 1.873e-02 3.018e-02 0.621 0.535115
## Saturationgray:I(R^2) 3.507e-01 4.509e-01 0.778 0.437004
## Saturationmuted:I(R^2) 4.412e-02 1.405e-02 3.140 0.001768 **
## Saturationneutral:I(R^2) -9.845e-02 1.477e-01 -0.667 0.505320
## Saturationpure:I(R^2) 8.689e-03 1.047e-02 0.830 0.406918
## Saturationshaded:I(R^2) -9.684e-02 4.802e-02 -2.016 0.044166 *
## Saturationsubdued:I(R^2) 4.059e-02 2.604e-02 1.559 0.119496
## Saturationgray:I(G^2) 5.604e-02 6.991e-01 0.080 0.936140
## Saturationmuted:I(G^2) 5.587e-02 3.474e-02 1.608 0.108240
## Saturationneutral:I(G^2) -4.113e-01 2.176e-01 -1.890 0.059155 .
## Saturationpure:I(G^2) 1.592e-02 2.779e-02 0.573 0.566951
## Saturationshaded:I(G^2) -6.115e-02 9.074e-02 -0.674 0.500651
## Saturationsubdued:I(G^2) -1.946e-02 4.496e-02 -0.433 0.665365
## Saturationgray:I(B^2) -2.276e-01 2.326e-01 -0.979 0.328175
## Saturationmuted:I(B^2) 5.149e-02 2.271e-02 2.267 0.023712 *
## Saturationneutral:I(B^2) -1.073e-01 7.824e-02 -1.371 0.170714
## Saturationpure:I(B^2) -1.393e-02 1.544e-02 -0.902 0.367315
## Saturationshaded:I(B^2) 4.691e-02 4.523e-02 1.037 0.300067
## Saturationsubdued:I(B^2) 8.180e-03 2.649e-02 0.309 0.757579
## Saturationgray:I(Hue^2) -1.191e-03 1.931e-02 -0.062 0.950860
## Saturationmuted:I(Hue^2) -2.269e-02 1.532e-02 -1.481 0.139129
## Saturationneutral:I(Hue^2) 1.765e-02 1.625e-02 1.086 0.277735
## Saturationpure:I(Hue^2) 1.881e-02 1.399e-02 1.344 0.179297
## Saturationshaded:I(Hue^2) 1.850e-03 1.498e-02 0.124 0.901716
## Saturationsubdued:I(Hue^2) -1.052e-02 1.565e-02 -0.672 0.501866
## Lightnessdeep:R:G 5.004e-03 1.886e-02 0.265 0.790829
## Lightnesslight:R:G 2.533e-01 1.325e-01 1.912 0.056313 .
## Lightnessmidtone:R:G 1.229e-01 4.718e-02 2.605 0.009400 **
## Lightnesspale:R:G 6.957e-01 3.645e-01 1.909 0.056737 .
## Lightnesssaturated:R:G 7.351e-02 2.793e-02 2.632 0.008697 **
## Lightnesssoft:R:G 1.012e-01 7.566e-02 1.337 0.181688
## Lightnessdeep:R:B -1.165e-02 2.386e-02 -0.488 0.625555
## Lightnesslight:R:B -6.598e-02 1.026e-01 -0.643 0.520627
## Lightnessmidtone:R:B 4.224e-03 3.560e-02 0.119 0.905599
## Lightnesspale:R:B -4.629e-01 2.799e-01 -1.654 0.098693 .
## Lightnesssaturated:R:B 4.583e-02 2.789e-02 1.643 0.100854
## Lightnesssoft:R:B 8.404e-03 5.500e-02 0.153 0.878610
## Lightnessdeep:R:Hue -3.360e-03 1.729e-02 -0.194 0.845981
## Lightnesslight:R:Hue 2.510e-03 4.470e-02 0.056 0.955240
## Lightnessmidtone:R:Hue -6.270e-03 2.600e-02 -0.241 0.809552
## Lightnesspale:R:Hue 1.245e-01 7.858e-02 1.585 0.113513
## Lightnesssaturated:R:Hue -1.019e-02 2.032e-02 -0.501 0.616402
## Lightnesssoft:R:Hue -1.562e-03 3.446e-02 -0.045 0.963867
## Lightnessdeep:G:B 1.105e-01 3.088e-02 3.577 0.000373 ***
## Lightnesslight:G:B 2.217e-01 1.222e-01 1.815 0.070063 .
## Lightnessmidtone:G:B 1.522e-01 4.758e-02 3.198 0.001450 **
## Lightnesspale:G:B 5.919e-01 3.194e-01 1.853 0.064293 .
## Lightnesssaturated:G:B 1.181e-01 3.608e-02 3.272 0.001127 **
## Lightnesssoft:G:B 1.260e-01 7.096e-02 1.776 0.076212 .
## Lightnessdeep:G:Hue -8.602e-02 3.038e-02 -2.831 0.004787 **
## Lightnesslight:G:Hue -3.120e-03 5.601e-02 -0.056 0.955598
## Lightnessmidtone:G:Hue -1.028e-02 4.171e-02 -0.247 0.805323
## Lightnesspale:G:Hue -4.457e-02 7.640e-02 -0.583 0.559832
## Lightnesssaturated:G:Hue -2.276e-02 3.461e-02 -0.658 0.511073
## Lightnesssoft:G:Hue -4.206e-02 4.633e-02 -0.908 0.364315
## Lightnessdeep:B:Hue -8.372e-02 2.929e-02 -2.858 0.004397 **
## Lightnesslight:B:Hue -1.382e-01 4.860e-02 -2.844 0.004604 **
## Lightnessmidtone:B:Hue -8.364e-02 3.525e-02 -2.373 0.017953 *
## Lightnesspale:B:Hue -1.631e-01 6.742e-02 -2.418 0.015863 *
## Lightnesssaturated:B:Hue -5.887e-02 3.048e-02 -1.931 0.053880 .
## Lightnesssoft:B:Hue -8.644e-02 4.071e-02 -2.123 0.034103 *
## Saturationgray:R:G -7.067e-01 1.068e+00 -0.662 0.508266
## Saturationmuted:R:G -1.153e-01 3.770e-02 -3.059 0.002316 **
## Saturationneutral:R:G 3.978e-01 3.232e-01 1.231 0.218819
## Saturationpure:R:G 3.260e-06 3.163e-02 0.000 0.999918
## Saturationshaded:R:G 1.796e-01 1.030e-01 1.743 0.081838 .
## Saturationsubdued:R:G -7.833e-02 5.576e-02 -1.405 0.160537
## Saturationgray:R:B 2.204e-01 4.825e-01 0.457 0.647921
## Saturationmuted:R:B 5.159e-02 3.079e-02 1.676 0.094308 .
## Saturationneutral:R:B -2.422e-01 1.320e-01 -1.836 0.066895 .
## Saturationpure:R:B -1.737e-02 2.142e-02 -0.811 0.417748
## Saturationshaded:R:B -4.993e-02 7.533e-02 -0.663 0.507664
## Saturationsubdued:R:B 1.566e-02 3.860e-02 0.406 0.684978
## Saturationgray:R:Hue 5.358e-02 1.348e-01 0.397 0.691182
## Saturationmuted:R:Hue -8.457e-03 3.214e-02 -0.263 0.792529
## Saturationneutral:R:Hue 3.524e-02 7.918e-02 0.445 0.656472
## Saturationpure:R:Hue -2.235e-02 3.801e-02 -0.588 0.556769
## Saturationshaded:R:Hue -4.133e-02 4.705e-02 -0.878 0.380105
## Saturationsubdued:R:Hue -2.503e-02 3.787e-02 -0.661 0.508859
## Saturationgray:G:B 3.230e-01 6.893e-01 0.469 0.639517
## Saturationmuted:G:B -7.810e-02 5.265e-02 -1.483 0.138482
## Saturationneutral:G:B 4.554e-01 2.291e-01 1.987 0.047300 *
## Saturationpure:G:B 1.353e-02 3.454e-02 0.392 0.695318
## Saturationshaded:G:B -3.276e-02 1.269e-01 -0.258 0.796315
## Saturationsubdued:G:B 4.067e-02 6.306e-02 0.645 0.519252
## Saturationgray:G:Hue -1.215e-01 1.564e-01 -0.777 0.437700
## Saturationmuted:G:Hue -1.850e-02 2.909e-02 -0.636 0.525012
## Saturationneutral:G:Hue -1.792e-02 7.618e-02 -0.235 0.814089
## Saturationpure:G:Hue 9.135e-03 2.725e-02 0.335 0.737552
## Saturationshaded:G:Hue 7.314e-02 4.226e-02 1.731 0.083967 .
## Saturationsubdued:G:Hue -3.253e-02 3.569e-02 -0.911 0.362429
## Saturationgray:B:Hue 7.069e-02 9.546e-02 0.741 0.459213
## Saturationmuted:B:Hue 1.980e-02 3.904e-02 0.507 0.612201
## Saturationneutral:B:Hue -4.266e-02 5.169e-02 -0.825 0.409520
## Saturationpure:B:Hue -7.908e-03 3.794e-02 -0.208 0.834956
## Saturationshaded:B:Hue -5.420e-02 4.629e-02 -1.171 0.242119
## Saturationsubdued:B:Hue 2.367e-02 4.151e-02 0.570 0.568665
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.03175 on 640 degrees of freedom
## Multiple R-squared: 0.9992, Adjusted R-squared: 0.999
## F-statistic: 4260 on 194 and 640 DF, p-value: < 2.2e-16#A9. Can consider interactions of basis functions with other basis functions!
fit_lm_09 <- lm(y ~ Lightness + Saturation + R*G*B*Hue + I(R^2) + I(G^2) + I(B^2) + I(Hue^2), data = df_standard)
fit_lm_09 %>% summary()##
## Call:
## lm(formula = y ~ Lightness + Saturation + R * G * B * Hue + I(R^2) +
## I(G^2) + I(B^2) + I(Hue^2), data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.197582 -0.027025 -0.002384 0.026201 0.250074
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.1657261 0.0174878 -9.477 < 2e-16 ***
## Lightnessdeep 0.0353333 0.0083990 4.207 2.88e-05 ***
## Lightnesslight 0.0058840 0.0192118 0.306 0.759480
## Lightnessmidtone 0.0456371 0.0161270 2.830 0.004773 **
## Lightnesspale 0.0573979 0.0199970 2.870 0.004208 **
## Lightnesssaturated 0.0502504 0.0125656 3.999 6.95e-05 ***
## Lightnesssoft 0.0190397 0.0179929 1.058 0.290291
## Saturationgray 0.0054304 0.0082281 0.660 0.509455
## Saturationmuted 0.0081681 0.0059433 1.374 0.169723
## Saturationneutral -0.0018902 0.0068573 -0.276 0.782886
## Saturationpure 0.0217553 0.0063061 3.450 0.000590 ***
## Saturationshaded 0.0036000 0.0065442 0.550 0.582405
## Saturationsubdued 0.0007582 0.0062741 0.121 0.903848
## R 0.2396113 0.0073988 32.385 < 2e-16 ***
## G 0.6455037 0.0106439 60.645 < 2e-16 ***
## B 0.1285391 0.0097490 13.185 < 2e-16 ***
## Hue 0.0129430 0.0057774 2.240 0.025346 *
## I(R^2) 0.0490690 0.0042808 11.463 < 2e-16 ***
## I(G^2) 0.0957508 0.0078753 12.158 < 2e-16 ***
## I(B^2) 0.0396873 0.0059323 6.690 4.18e-11 ***
## I(Hue^2) -0.0162166 0.0033058 -4.906 1.13e-06 ***
## R:G 0.0177352 0.0085847 2.066 0.039157 *
## R:B -0.0051333 0.0073410 -0.699 0.484585
## G:B -0.0123909 0.0100415 -1.234 0.217575
## R:Hue 0.0076914 0.0086092 0.893 0.371914
## G:Hue 0.0245189 0.0071463 3.431 0.000632 ***
## B:Hue -0.0318680 0.0096608 -3.299 0.001014 **
## R:G:B 0.0994143 0.0043243 22.990 < 2e-16 ***
## R:G:Hue -0.0476393 0.0067120 -7.098 2.79e-12 ***
## R:B:Hue 0.0116556 0.0082037 1.421 0.155770
## G:B:Hue 0.0313149 0.0051365 6.097 1.68e-09 ***
## R:G:B:Hue -0.0055666 0.0032905 -1.692 0.091092 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.04481 on 803 degrees of freedom
## Multiple R-squared: 0.9981, Adjusted R-squared: 0.998
## F-statistic: 1.338e+04 on 31 and 803 DF, p-value: < 2.2e-16#A10. Can consider interactions of basis functions with the categorical inputs!
fit_lm_10 <- lm(y ~ (Lightness + Saturation) * R*G*B*Hue, data = df_standard)##save model
fit_lm_01 %>% readr::write_rds("mod01.rds")
fit_lm_02 %>% readr::write_rds("mod02.rds")
fit_lm_03 %>% readr::write_rds("mod03.rds")
fit_lm_04 %>% readr::write_rds("mod04.rds")
fit_lm_05 %>% readr::write_rds("mod05.rds")
fit_lm_06 %>% readr::write_rds("mod06.rds")
fit_lm_07 %>% readr::write_rds("mod07.rds")
fit_lm_08 %>% readr::write_rds("mod08.rds")
fit_lm_09 %>% readr::write_rds("mod09.rds")
fit_lm_10 %>% readr::write_rds("mod10.rds")##reload model
re_load_mod01 <- readr::read_rds("mod01.rds")
re_load_mod02 <- readr::read_rds("mod02.rds")
re_load_mod03 <- readr::read_rds("mod03.rds")
re_load_mod04 <- readr::read_rds("mod04.rds")
re_load_mod05 <- readr::read_rds("mod05.rds")
re_load_mod06 <- readr::read_rds("mod06.rds")
re_load_mod07 <- readr::read_rds("mod07.rds")
re_load_mod08 <- readr::read_rds("mod08.rds")
re_load_mod09 <- readr::read_rds("mod09.rds")
re_load_mod10 <- readr::read_rds("mod10.rds")Part ii: Regression - iiA) Linear models
#1. Which of the 10 models is the best? What performance metric did you use to make your selection?
extract_metrics <- function(mod, mod_name)
{
broom::glance(mod) %>% mutate(mod_name = mod_name)
}
all_metrics <- purrr::map2_dfr(list(re_load_mod01, re_load_mod02, re_load_mod03, re_load_mod04, re_load_mod05, re_load_mod06, re_load_mod07, re_load_mod08, re_load_mod09, re_load_mod10),
sprintf("%02d", 1:10),
extract_metrics)
all_metrics %>% glimpse()## Rows: 10
## Columns: 13
## $ r.squared <dbl> 0.0000000, 0.8846326, 0.9881038, 0.9945082, 0.9978083, 0…
## $ adj.r.squared <dbl> 0.0000000, 0.8829484, 0.9880465, 0.9944008, 0.9976262, 0…
## $ sigma <dbl> 1.00000000, 0.34212801, 0.10933222, 0.07482803, 0.048722…
## $ statistic <dbl> NA, 525.2554, 17235.0403, 9258.1844, 5477.4752, 8494.316…
## $ p.value <dbl> NA, 0, 0, 0, 0, 0, 0, 0, 0, 0
## $ df <dbl> NA, 12, 4, 16, 64, 22, 142, 194, 31, 207
## $ logLik <dbl> -1184.3134, -282.6663, 665.8529, 988.5639, 1372.0751, 10…
## $ AIC <dbl> 2372.6268, 593.3327, -1319.7058, -1941.1279, -2612.1502,…
## $ BIC <dbl> 2382.0816, 659.5167, -1291.3412, -1856.0341, -2300.1397,…
## $ deviance <dbl> 834.0000000, 96.2163935, 9.9214333, 4.5801732, 1.8278624…
## $ df.residual <int> 834, 822, 830, 818, 770, 812, 692, 640, 803, 627
## $ nobs <int> 835, 835, 835, 835, 835, 835, 835, 835, 835, 835
## $ mod_name <chr> "01", "02", "03", "04", "05", "06", "07", "08", "09", "1…#2. Visualize the coefficient summaries for your top 3 models.
all_metrics %>%
select(mod_name, AIC, BIC) %>%
pivot_longer(c(AIC, BIC)) %>%
ggplot(mapping = aes(x = mod_name, y = value)) +
geom_point(size = 2) +
facet_wrap(~name, scales = 'free_y') +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
Based on the information provided, model 8 is selected as the optimal model by the less conservative Akaike Information Criterion (AIC), while the more conservative Bayesian Information Criterion (BIC) indicates that model 9 is preferable. Given that my evaluation criteria prioritize BIC as the performance metric, I conclude that model 9 is the superior model.
#3. How do the coefficient summaries compare between the top 3 models?
Model 9 considers the smallest set of features, whereas model 8 incorporates the broadest range. Both models concur on the significance of certain features associated with G.
• Which inputs seem important? In examining the summaries of models 9, 8, and 5, it appears that the features related to the continuous variable G are of significance.
re_load_mod09 %>% coefplot::coefplot()
re_load_mod08 %>% coefplot::coefplot()
re_load_mod05 %>% coefplot::coefplot()
re_load_mod09 %>% summary()##
## Call:
## lm(formula = y ~ Lightness + Saturation + R * G * B * Hue + I(R^2) +
## I(G^2) + I(B^2) + I(Hue^2), data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.197582 -0.027025 -0.002384 0.026201 0.250074
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.1657261 0.0174878 -9.477 < 2e-16 ***
## Lightnessdeep 0.0353333 0.0083990 4.207 2.88e-05 ***
## Lightnesslight 0.0058840 0.0192118 0.306 0.759480
## Lightnessmidtone 0.0456371 0.0161270 2.830 0.004773 **
## Lightnesspale 0.0573979 0.0199970 2.870 0.004208 **
## Lightnesssaturated 0.0502504 0.0125656 3.999 6.95e-05 ***
## Lightnesssoft 0.0190397 0.0179929 1.058 0.290291
## Saturationgray 0.0054304 0.0082281 0.660 0.509455
## Saturationmuted 0.0081681 0.0059433 1.374 0.169723
## Saturationneutral -0.0018902 0.0068573 -0.276 0.782886
## Saturationpure 0.0217553 0.0063061 3.450 0.000590 ***
## Saturationshaded 0.0036000 0.0065442 0.550 0.582405
## Saturationsubdued 0.0007582 0.0062741 0.121 0.903848
## R 0.2396113 0.0073988 32.385 < 2e-16 ***
## G 0.6455037 0.0106439 60.645 < 2e-16 ***
## B 0.1285391 0.0097490 13.185 < 2e-16 ***
## Hue 0.0129430 0.0057774 2.240 0.025346 *
## I(R^2) 0.0490690 0.0042808 11.463 < 2e-16 ***
## I(G^2) 0.0957508 0.0078753 12.158 < 2e-16 ***
## I(B^2) 0.0396873 0.0059323 6.690 4.18e-11 ***
## I(Hue^2) -0.0162166 0.0033058 -4.906 1.13e-06 ***
## R:G 0.0177352 0.0085847 2.066 0.039157 *
## R:B -0.0051333 0.0073410 -0.699 0.484585
## G:B -0.0123909 0.0100415 -1.234 0.217575
## R:Hue 0.0076914 0.0086092 0.893 0.371914
## G:Hue 0.0245189 0.0071463 3.431 0.000632 ***
## B:Hue -0.0318680 0.0096608 -3.299 0.001014 **
## R:G:B 0.0994143 0.0043243 22.990 < 2e-16 ***
## R:G:Hue -0.0476393 0.0067120 -7.098 2.79e-12 ***
## R:B:Hue 0.0116556 0.0082037 1.421 0.155770
## G:B:Hue 0.0313149 0.0051365 6.097 1.68e-09 ***
## R:G:B:Hue -0.0055666 0.0032905 -1.692 0.091092 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.04481 on 803 degrees of freedom
## Multiple R-squared: 0.9981, Adjusted R-squared: 0.998
## F-statistic: 1.338e+04 on 31 and 803 DF, p-value: < 2.2e-16re_load_mod08 %>% summary()##
## Call:
## lm(formula = y ~ (Lightness + Saturation) * ((R + G + B + Hue)^2 +
## I(R^2) + I(G^2) + I(B^2) + I(Hue^2)), data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.125500 -0.015352 -0.000605 0.015508 0.133380
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.740e-01 1.229e-01 -4.669 3.68e-06 ***
## Lightnessdeep 4.347e-01 1.180e-01 3.683 0.000250 ***
## Lightnesslight 5.684e-01 1.325e-01 4.289 2.07e-05 ***
## Lightnessmidtone 4.396e-01 1.242e-01 3.539 0.000430 ***
## Lightnesspale 1.024e+00 1.769e-01 5.790 1.10e-08 ***
## Lightnesssaturated 4.619e-01 1.207e-01 3.827 0.000142 ***
## Lightnesssoft 4.665e-01 1.248e-01 3.738 0.000202 ***
## Saturationgray -1.993e-02 3.240e-02 -0.615 0.538765
## Saturationmuted -1.097e-02 3.098e-02 -0.354 0.723496
## Saturationneutral -2.988e-03 3.065e-02 -0.098 0.922355
## Saturationpure -5.557e-02 4.475e-02 -1.242 0.214840
## Saturationshaded -6.139e-04 3.087e-02 -0.020 0.984140
## Saturationsubdued -1.784e-02 3.093e-02 -0.577 0.564358
## R 1.631e-01 4.856e-02 3.359 0.000829 ***
## G 4.170e-01 5.547e-02 7.517 1.90e-13 ***
## B -4.034e-01 1.302e-01 -3.097 0.002038 **
## Hue 1.331e-01 6.460e-02 2.060 0.039763 *
## I(R^2) 4.656e-02 9.126e-03 5.102 4.44e-07 ***
## I(G^2) 2.743e-02 2.873e-02 0.955 0.340111
## I(B^2) -1.243e-01 3.887e-02 -3.197 0.001456 **
## I(Hue^2) 2.758e-03 1.805e-02 0.153 0.878584
## R:G -8.968e-02 3.068e-02 -2.923 0.003584 **
## R:B -2.549e-02 2.682e-02 -0.950 0.342234
## R:Hue 2.807e-02 3.164e-02 0.887 0.375308
## G:B -1.531e-01 4.267e-02 -3.589 0.000358 ***
## G:Hue 3.767e-02 3.407e-02 1.105 0.269365
## B:Hue 8.085e-02 3.987e-02 2.028 0.043005 *
## Lightnessdeep:R -7.683e-03 4.321e-02 -0.178 0.858925
## Lightnesslight:R 1.451e-01 1.371e-01 1.058 0.290250
## Lightnessmidtone:R 5.111e-02 4.755e-02 1.075 0.282878
## Lightnesspale:R -3.972e-01 3.646e-01 -1.089 0.276491
## Lightnesssaturated:R 6.853e-02 4.376e-02 1.566 0.117829
## Lightnesssoft:R 5.585e-02 6.214e-02 0.899 0.369108
## Lightnessdeep:G 2.248e-01 5.025e-02 4.473 9.12e-06 ***
## Lightnesslight:G -2.549e-01 1.660e-01 -1.536 0.125121
## Lightnessmidtone:G 2.251e-01 5.900e-02 3.815 0.000150 ***
## Lightnesspale:G -8.384e-01 4.563e-01 -1.837 0.066623 .
## Lightnesssaturated:G 2.284e-01 5.160e-02 4.427 1.13e-05 ***
## Lightnesssoft:G 7.211e-02 8.030e-02 0.898 0.369491
## Lightnessdeep:B 5.033e-01 1.260e-01 3.996 7.18e-05 ***
## Lightnesslight:B 4.837e-01 1.612e-01 3.001 0.002799 **
## Lightnessmidtone:B 5.347e-01 1.315e-01 4.067 5.36e-05 ***
## Lightnesspale:B 4.947e-01 2.378e-01 2.080 0.037898 *
## Lightnesssaturated:B 5.630e-01 1.260e-01 4.468 9.34e-06 ***
## Lightnesssoft:B 5.238e-01 1.354e-01 3.870 0.000120 ***
## Lightnessdeep:Hue -2.048e-01 6.599e-02 -3.104 0.001994 **
## Lightnesslight:Hue -1.332e-01 7.253e-02 -1.837 0.066663 .
## Lightnessmidtone:Hue -1.589e-01 6.363e-02 -2.497 0.012772 *
## Lightnesspale:Hue -1.625e-01 8.888e-02 -1.828 0.068054 .
## Lightnesssaturated:Hue -1.444e-01 6.403e-02 -2.255 0.024445 *
## Lightnesssoft:Hue -1.594e-01 6.563e-02 -2.429 0.015411 *
## Lightnessdeep:I(R^2) -9.327e-03 8.874e-03 -1.051 0.293647
## Lightnesslight:I(R^2) -7.219e-02 7.377e-02 -0.979 0.328193
## Lightnessmidtone:I(R^2) 4.760e-03 1.779e-02 0.268 0.789132
## Lightnesspale:I(R^2) 2.988e-01 2.231e-01 1.339 0.180897
## Lightnesssaturated:I(R^2) 1.755e-03 9.818e-03 0.179 0.858179
## Lightnesssoft:I(R^2) 1.044e-02 3.702e-02 0.282 0.778111
## Lightnessdeep:I(G^2) 6.690e-02 2.053e-02 3.258 0.001180 **
## Lightnesslight:I(G^2) 3.432e-01 1.293e-01 2.654 0.008158 **
## Lightnessmidtone:I(G^2) 9.108e-02 4.472e-02 2.037 0.042087 *
## Lightnesspale:I(G^2) 2.753e-01 3.361e-01 0.819 0.413091
## Lightnesssaturated:I(G^2) 6.074e-02 2.885e-02 2.105 0.035667 *
## Lightnesssoft:I(G^2) 2.548e-01 7.069e-02 3.605 0.000337 ***
## Lightnessdeep:I(B^2) 1.410e-01 3.725e-02 3.785 0.000168 ***
## Lightnesslight:I(B^2) 2.097e-01 6.587e-02 3.184 0.001523 **
## Lightnessmidtone:I(B^2) 1.675e-01 3.893e-02 4.304 1.94e-05 ***
## Lightnesspale:I(B^2) 2.031e-01 1.088e-01 1.867 0.062397 .
## Lightnesssaturated:I(B^2) 1.667e-01 3.840e-02 4.341 1.65e-05 ***
## Lightnesssoft:I(B^2) 2.116e-01 4.521e-02 4.681 3.49e-06 ***
## Lightnessdeep:I(Hue^2) -3.804e-03 1.196e-02 -0.318 0.750587
## Lightnesslight:I(Hue^2) -4.958e-03 1.680e-02 -0.295 0.768033
## Lightnessmidtone:I(Hue^2) -6.729e-03 1.573e-02 -0.428 0.668964
## Lightnesspale:I(Hue^2) -4.647e-03 1.597e-02 -0.291 0.771202
## Lightnesssaturated:I(Hue^2) -1.369e-02 1.332e-02 -1.028 0.304457
## Lightnesssoft:I(Hue^2) -1.799e-02 1.718e-02 -1.047 0.295356
## Saturationgray:R -7.726e-03 1.336e-01 -0.058 0.953893
## Saturationmuted:R 1.801e-03 2.805e-02 0.064 0.948832
## Saturationneutral:R 6.425e-02 5.264e-02 1.221 0.222646
## Saturationpure:R 3.889e-02 3.024e-02 1.286 0.198924
## Saturationshaded:R -8.517e-03 3.989e-02 -0.214 0.830987
## Saturationsubdued:R 2.550e-02 3.297e-02 0.773 0.439627
## Saturationgray:G -1.178e-01 1.631e-01 -0.723 0.470079
## Saturationmuted:G 6.606e-04 4.410e-02 0.015 0.988053
## Saturationneutral:G -6.066e-02 8.990e-02 -0.675 0.500036
## Saturationpure:G 4.224e-02 3.922e-02 1.077 0.281977
## Saturationshaded:G 3.322e-02 6.555e-02 0.507 0.612488
## Saturationsubdued:G -4.625e-04 5.330e-02 -0.009 0.993079
## Saturationgray:B 1.423e-01 1.037e-01 1.372 0.170585
## Saturationmuted:B 2.640e-02 3.160e-02 0.835 0.403833
## Saturationneutral:B 1.006e-02 5.894e-02 0.171 0.864533
## Saturationpure:B -3.860e-02 2.836e-02 -1.361 0.174060
## Saturationshaded:B -2.382e-02 4.371e-02 -0.545 0.585942
## Saturationsubdued:B 7.099e-04 3.715e-02 0.019 0.984760
## Saturationgray:Hue 3.804e-05 3.719e-02 0.001 0.999184
## Saturationmuted:Hue 2.625e-04 2.964e-02 0.009 0.992935
## Saturationneutral:Hue 4.235e-02 3.060e-02 1.384 0.166833
## Saturationpure:Hue 3.863e-02 3.700e-02 1.044 0.296781
## Saturationshaded:Hue 2.487e-02 3.019e-02 0.824 0.410388
## Saturationsubdued:Hue 1.873e-02 3.018e-02 0.621 0.535115
## Saturationgray:I(R^2) 3.507e-01 4.509e-01 0.778 0.437004
## Saturationmuted:I(R^2) 4.412e-02 1.405e-02 3.140 0.001768 **
## Saturationneutral:I(R^2) -9.845e-02 1.477e-01 -0.667 0.505320
## Saturationpure:I(R^2) 8.689e-03 1.047e-02 0.830 0.406918
## Saturationshaded:I(R^2) -9.684e-02 4.802e-02 -2.016 0.044166 *
## Saturationsubdued:I(R^2) 4.059e-02 2.604e-02 1.559 0.119496
## Saturationgray:I(G^2) 5.604e-02 6.991e-01 0.080 0.936140
## Saturationmuted:I(G^2) 5.587e-02 3.474e-02 1.608 0.108240
## Saturationneutral:I(G^2) -4.113e-01 2.176e-01 -1.890 0.059155 .
## Saturationpure:I(G^2) 1.592e-02 2.779e-02 0.573 0.566951
## Saturationshaded:I(G^2) -6.115e-02 9.074e-02 -0.674 0.500651
## Saturationsubdued:I(G^2) -1.946e-02 4.496e-02 -0.433 0.665365
## Saturationgray:I(B^2) -2.276e-01 2.326e-01 -0.979 0.328175
## Saturationmuted:I(B^2) 5.149e-02 2.271e-02 2.267 0.023712 *
## Saturationneutral:I(B^2) -1.073e-01 7.824e-02 -1.371 0.170714
## Saturationpure:I(B^2) -1.393e-02 1.544e-02 -0.902 0.367315
## Saturationshaded:I(B^2) 4.691e-02 4.523e-02 1.037 0.300067
## Saturationsubdued:I(B^2) 8.180e-03 2.649e-02 0.309 0.757579
## Saturationgray:I(Hue^2) -1.191e-03 1.931e-02 -0.062 0.950860
## Saturationmuted:I(Hue^2) -2.269e-02 1.532e-02 -1.481 0.139129
## Saturationneutral:I(Hue^2) 1.765e-02 1.625e-02 1.086 0.277735
## Saturationpure:I(Hue^2) 1.881e-02 1.399e-02 1.344 0.179297
## Saturationshaded:I(Hue^2) 1.850e-03 1.498e-02 0.124 0.901716
## Saturationsubdued:I(Hue^2) -1.052e-02 1.565e-02 -0.672 0.501866
## Lightnessdeep:R:G 5.004e-03 1.886e-02 0.265 0.790829
## Lightnesslight:R:G 2.533e-01 1.325e-01 1.912 0.056313 .
## Lightnessmidtone:R:G 1.229e-01 4.718e-02 2.605 0.009400 **
## Lightnesspale:R:G 6.957e-01 3.645e-01 1.909 0.056737 .
## Lightnesssaturated:R:G 7.351e-02 2.793e-02 2.632 0.008697 **
## Lightnesssoft:R:G 1.012e-01 7.566e-02 1.337 0.181688
## Lightnessdeep:R:B -1.165e-02 2.386e-02 -0.488 0.625555
## Lightnesslight:R:B -6.598e-02 1.026e-01 -0.643 0.520627
## Lightnessmidtone:R:B 4.224e-03 3.560e-02 0.119 0.905599
## Lightnesspale:R:B -4.629e-01 2.799e-01 -1.654 0.098693 .
## Lightnesssaturated:R:B 4.583e-02 2.789e-02 1.643 0.100854
## Lightnesssoft:R:B 8.404e-03 5.500e-02 0.153 0.878610
## Lightnessdeep:R:Hue -3.360e-03 1.729e-02 -0.194 0.845981
## Lightnesslight:R:Hue 2.510e-03 4.470e-02 0.056 0.955240
## Lightnessmidtone:R:Hue -6.270e-03 2.600e-02 -0.241 0.809552
## Lightnesspale:R:Hue 1.245e-01 7.858e-02 1.585 0.113513
## Lightnesssaturated:R:Hue -1.019e-02 2.032e-02 -0.501 0.616402
## Lightnesssoft:R:Hue -1.562e-03 3.446e-02 -0.045 0.963867
## Lightnessdeep:G:B 1.105e-01 3.088e-02 3.577 0.000373 ***
## Lightnesslight:G:B 2.217e-01 1.222e-01 1.815 0.070063 .
## Lightnessmidtone:G:B 1.522e-01 4.758e-02 3.198 0.001450 **
## Lightnesspale:G:B 5.919e-01 3.194e-01 1.853 0.064293 .
## Lightnesssaturated:G:B 1.181e-01 3.608e-02 3.272 0.001127 **
## Lightnesssoft:G:B 1.260e-01 7.096e-02 1.776 0.076212 .
## Lightnessdeep:G:Hue -8.602e-02 3.038e-02 -2.831 0.004787 **
## Lightnesslight:G:Hue -3.120e-03 5.601e-02 -0.056 0.955598
## Lightnessmidtone:G:Hue -1.028e-02 4.171e-02 -0.247 0.805323
## Lightnesspale:G:Hue -4.457e-02 7.640e-02 -0.583 0.559832
## Lightnesssaturated:G:Hue -2.276e-02 3.461e-02 -0.658 0.511073
## Lightnesssoft:G:Hue -4.206e-02 4.633e-02 -0.908 0.364315
## Lightnessdeep:B:Hue -8.372e-02 2.929e-02 -2.858 0.004397 **
## Lightnesslight:B:Hue -1.382e-01 4.860e-02 -2.844 0.004604 **
## Lightnessmidtone:B:Hue -8.364e-02 3.525e-02 -2.373 0.017953 *
## Lightnesspale:B:Hue -1.631e-01 6.742e-02 -2.418 0.015863 *
## Lightnesssaturated:B:Hue -5.887e-02 3.048e-02 -1.931 0.053880 .
## Lightnesssoft:B:Hue -8.644e-02 4.071e-02 -2.123 0.034103 *
## Saturationgray:R:G -7.067e-01 1.068e+00 -0.662 0.508266
## Saturationmuted:R:G -1.153e-01 3.770e-02 -3.059 0.002316 **
## Saturationneutral:R:G 3.978e-01 3.232e-01 1.231 0.218819
## Saturationpure:R:G 3.260e-06 3.163e-02 0.000 0.999918
## Saturationshaded:R:G 1.796e-01 1.030e-01 1.743 0.081838 .
## Saturationsubdued:R:G -7.833e-02 5.576e-02 -1.405 0.160537
## Saturationgray:R:B 2.204e-01 4.825e-01 0.457 0.647921
## Saturationmuted:R:B 5.159e-02 3.079e-02 1.676 0.094308 .
## Saturationneutral:R:B -2.422e-01 1.320e-01 -1.836 0.066895 .
## Saturationpure:R:B -1.737e-02 2.142e-02 -0.811 0.417748
## Saturationshaded:R:B -4.993e-02 7.533e-02 -0.663 0.507664
## Saturationsubdued:R:B 1.566e-02 3.860e-02 0.406 0.684978
## Saturationgray:R:Hue 5.358e-02 1.348e-01 0.397 0.691182
## Saturationmuted:R:Hue -8.457e-03 3.214e-02 -0.263 0.792529
## Saturationneutral:R:Hue 3.524e-02 7.918e-02 0.445 0.656472
## Saturationpure:R:Hue -2.235e-02 3.801e-02 -0.588 0.556769
## Saturationshaded:R:Hue -4.133e-02 4.705e-02 -0.878 0.380105
## Saturationsubdued:R:Hue -2.503e-02 3.787e-02 -0.661 0.508859
## Saturationgray:G:B 3.230e-01 6.893e-01 0.469 0.639517
## Saturationmuted:G:B -7.810e-02 5.265e-02 -1.483 0.138482
## Saturationneutral:G:B 4.554e-01 2.291e-01 1.987 0.047300 *
## Saturationpure:G:B 1.353e-02 3.454e-02 0.392 0.695318
## Saturationshaded:G:B -3.276e-02 1.269e-01 -0.258 0.796315
## Saturationsubdued:G:B 4.067e-02 6.306e-02 0.645 0.519252
## Saturationgray:G:Hue -1.215e-01 1.564e-01 -0.777 0.437700
## Saturationmuted:G:Hue -1.850e-02 2.909e-02 -0.636 0.525012
## Saturationneutral:G:Hue -1.792e-02 7.618e-02 -0.235 0.814089
## Saturationpure:G:Hue 9.135e-03 2.725e-02 0.335 0.737552
## Saturationshaded:G:Hue 7.314e-02 4.226e-02 1.731 0.083967 .
## Saturationsubdued:G:Hue -3.253e-02 3.569e-02 -0.911 0.362429
## Saturationgray:B:Hue 7.069e-02 9.546e-02 0.741 0.459213
## Saturationmuted:B:Hue 1.980e-02 3.904e-02 0.507 0.612201
## Saturationneutral:B:Hue -4.266e-02 5.169e-02 -0.825 0.409520
## Saturationpure:B:Hue -7.908e-03 3.794e-02 -0.208 0.834956
## Saturationshaded:B:Hue -5.420e-02 4.629e-02 -1.171 0.242119
## Saturationsubdued:B:Hue 2.367e-02 4.151e-02 0.570 0.568665
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.03175 on 640 degrees of freedom
## Multiple R-squared: 0.9992, Adjusted R-squared: 0.999
## F-statistic: 4260 on 194 and 640 DF, p-value: < 2.2e-16re_load_mod05 %>% summary()##
## Call:
## lm(formula = y ~ (Lightness + Saturation) * (R + G + B + Hue),
## data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.218807 -0.029587 -0.002964 0.023051 0.211793
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0156508 0.0333284 0.470 0.638778
## Lightnessdeep -0.0902948 0.0344015 -2.625 0.008844 **
## Lightnesslight -0.3303232 0.0419341 -7.877 1.14e-14 ***
## Lightnessmidtone -0.0813395 0.0341341 -2.383 0.017417 *
## Lightnesspale -0.6291915 0.0528630 -11.902 < 2e-16 ***
## Lightnesssaturated -0.0625005 0.0334789 -1.867 0.062300 .
## Lightnesssoft -0.1545354 0.0356907 -4.330 1.69e-05 ***
## Saturationgray -0.0440442 0.0115286 -3.820 0.000144 ***
## Saturationmuted -0.0291692 0.0063463 -4.596 5.03e-06 ***
## Saturationneutral -0.0618573 0.0067847 -9.117 < 2e-16 ***
## Saturationpure 0.0518827 0.0071504 7.256 9.75e-13 ***
## Saturationshaded -0.0531590 0.0064988 -8.180 1.17e-15 ***
## Saturationsubdued -0.0478015 0.0064215 -7.444 2.62e-13 ***
## R 0.1892266 0.0088042 21.493 < 2e-16 ***
## G 0.6312372 0.0116501 54.183 < 2e-16 ***
## B 0.1602681 0.0173716 9.226 < 2e-16 ***
## Hue -0.0573637 0.0094517 -6.069 2.02e-09 ***
## Lightnessdeep:R 0.0018104 0.0094027 0.193 0.847370
## Lightnesslight:R 0.1036261 0.0244211 4.243 2.47e-05 ***
## Lightnessmidtone:R 0.0151836 0.0133692 1.136 0.256430
## Lightnesspale:R 0.1812149 0.0433573 4.180 3.26e-05 ***
## Lightnesssaturated:R -0.0007776 0.0106664 -0.073 0.941904
## Lightnesssoft:R 0.0492488 0.0186494 2.641 0.008439 **
## Lightnessdeep:G -0.0580356 0.0116771 -4.970 8.25e-07 ***
## Lightnesslight:G 0.3380262 0.0295808 11.427 < 2e-16 ***
## Lightnessmidtone:G 0.0665761 0.0164872 4.038 5.93e-05 ***
## Lightnesspale:G 0.5858475 0.0457749 12.798 < 2e-16 ***
## Lightnesssaturated:G -0.0118009 0.0126143 -0.936 0.349815
## Lightnesssoft:G 0.2086321 0.0225868 9.237 < 2e-16 ***
## Lightnessdeep:B -0.0494322 0.0201075 -2.458 0.014175 *
## Lightnesslight:B -0.0043713 0.0248889 -0.176 0.860630
## Lightnessmidtone:B -0.0644335 0.0201605 -3.196 0.001450 **
## Lightnesspale:B 0.0233714 0.0316768 0.738 0.460855
## Lightnesssaturated:B -0.0469872 0.0202082 -2.325 0.020323 *
## Lightnesssoft:B -0.0545076 0.0210505 -2.589 0.009797 **
## Lightnessdeep:Hue 0.0392576 0.0101835 3.855 0.000125 ***
## Lightnesslight:Hue 0.0525404 0.0098568 5.330 1.29e-07 ***
## Lightnessmidtone:Hue 0.0473387 0.0102896 4.601 4.92e-06 ***
## Lightnesspale:Hue 0.0544998 0.0103466 5.267 1.80e-07 ***
## Lightnesssaturated:Hue 0.0375392 0.0099384 3.777 0.000171 ***
## Lightnesssoft:Hue 0.0534146 0.0103915 5.140 3.48e-07 ***
## Saturationgray:R -0.1580780 0.0675196 -2.341 0.019475 *
## Saturationmuted:R -0.0187794 0.0114883 -1.635 0.102531
## Saturationneutral:R 0.0608227 0.0285038 2.134 0.033171 *
## Saturationpure:R -0.0011462 0.0086715 -0.132 0.894881
## Saturationshaded:R 0.0127518 0.0200588 0.636 0.525149
## Saturationsubdued:R 0.0060838 0.0152335 0.399 0.689732
## Saturationgray:G 0.2688204 0.0762993 3.523 0.000451 ***
## Saturationmuted:G 0.0382669 0.0135997 2.814 0.005021 **
## Saturationneutral:G -0.0058799 0.0331360 -0.177 0.859204
## Saturationpure:G 0.0340289 0.0110360 3.083 0.002119 **
## Saturationshaded:G 0.0410349 0.0237407 1.728 0.084307 .
## Saturationsubdued:G 0.0330834 0.0176914 1.870 0.061859 .
## Saturationgray:B -0.0628830 0.0409159 -1.537 0.124733
## Saturationmuted:B 0.0008769 0.0100322 0.087 0.930373
## Saturationneutral:B 0.0171906 0.0194636 0.883 0.377396
## Saturationpure:B -0.0400486 0.0086671 -4.621 4.48e-06 ***
## Saturationshaded:B 0.0014420 0.0148811 0.097 0.922827
## Saturationsubdued:B 0.0057016 0.0118594 0.481 0.630819
## Saturationgray:Hue 0.0016934 0.0121588 0.139 0.889268
## Saturationmuted:Hue -0.0075430 0.0087080 -0.866 0.386646
## Saturationneutral:Hue 0.0313044 0.0097180 3.221 0.001330 **
## Saturationpure:Hue 0.0137725 0.0085005 1.620 0.105598
## Saturationshaded:Hue 0.0020142 0.0091012 0.221 0.824907
## Saturationsubdued:Hue 0.0027768 0.0094296 0.294 0.768477
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.04872 on 770 degrees of freedom
## Multiple R-squared: 0.9978, Adjusted R-squared: 0.9976
## F-statistic: 5477 on 64 and 770 DF, p-value: < 2.2e-16Part ii: Regression– iiB) Bayesian Linear models
You have explored the relationships; next you must consider the UNCERTAINTY on the residual error through Bayesian modeling techniques! #1.Fit 2 Bayesian linear models – one must be the best model from iiA) ： model 9 the second must be another model you fit in iiA): model 8 • State why you chose the second model. • You may use the Laplace Approximation approach we used in lecture and the homework assignments. • Alternatively, you may use rstanarm’s stan_lm() or stan_glm() function to fit full Bayesian linear models with syntax like R’s lm(). • Resources to help with rstanarm if you’re interested: • How to Use the rstanarm Package (r-project.org) • Estimating Regularized Linear Models with rstanarm (r-project.org) • Extra examples also provided on Canvas.
• After fitting the 2 models, you must identify the best model. • Which performance metric did you use to make your selection? • Visualize the regression coefficient posterior summary statistics for your best model. • For your best model: Study the posterior UNCERTAINTY on the likelihood noise (residual error), 𝜎. • How does the lm() maximum likelihood estimate (MLE) on 𝜎 relate to the posterior UNCERTAINTY on 𝜎? • Do you feel the posterior is precise or are we quite uncertain about 𝜎?
Xmat_08 <- model.matrix(re_load_mod08, dfii)
Xmat_09 <- model.matrix(re_load_mod09, dfii)Create information with prior The info_09 list corresponds to the information for model 9, while info_08 corresponds to the information for model 08. Specify the shared prior mean, mu_beta, to be 0, the shared prior standard deviation, tau_beta, as 2. The prior rate parameter on the noise, sigma_rate, is assigned to 1.
info_08 <- list(
yobs = df_standard$y,
design_matrix = Xmat_08,
mu_beta = 0,
tau_beta = 2,
sigma_rate = 1
)
info_09 <- list(
yobs = df_standard$y,
design_matrix = Xmat_09,
mu_beta = 0,
tau_beta = 2,
sigma_rate = 1
)Define the log-posterior function lm_logpost(). Use the log-transformation on σ, and so we will actually define the log-posterior in terms of the regression coefficients, β, and the unbounded noise parameter, φ=log[σ].
lm_logpost <- function(unknowns, my_info)
{
length_beta <- length(unknowns)-1
beta_v <- unknowns[1:length_beta] %>% as.vector()
lik_varphi <- unknowns[length(unknowns)]
lik_sigma <- exp(lik_varphi)
X <- my_info$design_matrix
# calculate the linear predictor
mu <- X %*% beta_v
# evaluate the log-likelihood
log_lik <- sum(dnorm(x = my_info$yobs, mean = mu, sd = lik_sigma, log = TRUE))
# evaluate the log-prior
log_prior_beta <- sum(dnorm(x = beta_v, mean = my_info$mu_beta, sd = my_info$tau_beta, log = TRUE))
log_prior_sigma <- dexp(x = lik_sigma, rate = my_info$sigma_rate, log = TRUE)
# add the mean trend prior and noise prior together
log_prior <- log_prior_beta + log_prior_sigma
# account for the transformation
log_derive_adjust <- lik_varphi
# sum together
return(log_lik + log_prior + log_derive_adjust)
}The my_laplace() function is below
my_laplace <- function(start_guess, logpost_func, ...)
{
fit <- optim(start_guess,
logpost_func,
gr = NULL,
...,
method = "BFGS",
hessian = TRUE,
control = list(fnscale = -1, maxit = 1001))
mode <- fit$par
post_var_matrix <- -solve(fit$hessian)
p <- length(mode) # number of unknown parameters
int <- p/2 * log(2*pi) + 0.5*log(det(post_var_matrix)) + logpost_func(mode, ...)
# package all of the results into a list
list(mode = mode,
var_matrix = post_var_matrix,
log_evidence = int,
converge = ifelse(fit$convergence == 0,
"YES",
"NO"),
iter_counts = as.numeric(fit$counts[1]))
}Fit the Bayesian linear model 08.
init_guess <- rep(0, ncol(Xmat_08)+1)
laplace_quad_08 <- my_laplace(init_guess, lm_logpost, info_08)
laplace_quad_08$converge## [1] "YES"laplace_quad_08$log_evidence## [1] -InfDisplay the posterior mode and posterior standard deviations
cat("posterior mode: ", laplace_quad_08$mode, "\n\n")## posterior mode: -0.5513629 0.4137779 0.5438713 0.4166352 0.9896795 0.4397414 0.4431632 -0.01954001 -0.01091312 -0.002595044 -0.05529535 -0.0005549269 -0.01760662 0.166577 0.4209329 -0.3788962 0.1283146 0.04685251 0.02766046 -0.1179428 0.001884472 -0.08980574 -0.02363128 0.0282098 -0.1501897 0.03664078 0.07847458 -0.01062842 0.1386229 0.04779992 -0.4103492 0.06526968 0.05186577 0.2204176 -0.2522191 0.2207841 -0.8000247 0.223685 0.06949775 0.481451 0.4586167 0.5103998 0.457466 0.5398372 0.4990376 -0.2002474 -0.1293847 -0.1541937 -0.1591516 -0.1400747 -0.1550027 -0.009295456 -0.07118169 0.005019286 0.2996579 0.001819671 0.01064443 0.06702035 0.3412107 0.09227306 0.2703786 0.06181998 0.2545549 0.1355801 0.2047849 0.1618887 0.2009575 0.161222 0.2058263 -0.003823479 -0.004469975 -0.006330811 -0.004175158 -0.01386251 -0.01743817 -0.01915716 0.002064728 0.06427014 0.03872306 -0.008951481 0.02518748 -0.09967199 0.0004209751 -0.0617041 0.04179359 0.03413876 -0.0001129228 0.133334 0.02621953 0.011243 -0.0380661 -0.02441206 0.0004063613 -0.0001531142 0.0006291646 0.04206827 0.03795517 0.02495524 0.0187025 0.2865317 0.04393491 -0.08814013 0.008547119 -0.09753258 0.04014154 -0.02129725 0.05561047 -0.3963846 0.01603608 -0.06002794 -0.01903479 -0.228901 0.05114366 -0.1041692 -0.01339706 0.04719936 0.007861356 -0.0006670487 -0.02244184 0.01796383 0.01835175 0.002328396 -0.0100891 0.004484463 0.2524113 0.1213733 0.6786898 0.07215723 0.1005221 -0.01312053 -0.06498183 0.003397747 -0.4410236 0.04432404 0.007837539 -0.003116209 0.001980745 -0.006511085 0.122212 -0.01023658 -0.002219608 0.1076572 0.2152946 0.1481963 0.573516 0.1141759 0.1218802 -0.08552693 -0.001482859 -0.01005069 -0.0441347 -0.02260677 -0.04094213 -0.08129615 -0.1350991 -0.08065662 -0.1572569 -0.05638528 -0.08324537 -0.5585624 -0.1146378 0.375611 0.0002207609 0.1805006 -0.07772967 0.1749691 0.05110805 -0.2373467 -0.01693302 -0.05041805 0.01552658 0.0397311 -0.00873217 0.03991291 -0.02223833 -0.04172893 -0.02496097 0.3599958 -0.07757206 0.4439821 0.01259088 -0.03439237 0.04017654 -0.1032538 -0.01785263 -0.02215065 0.00944382 0.07389894 -0.03230595 0.06372182 0.01917394 -0.04205479 -0.00761598 -0.05475638 0.02338721 -3.582063cat("posterior standard deviations: ", sqrt(diag(laplace_quad_08$var_matrix)))## posterior standard deviations: 0.1055191 0.1013263 0.1138046 0.106519 0.1520288 0.1035246 0.1070151 0.0282161 0.02708923 0.02677497 0.03914197 0.02698191 0.02703662 0.04224551 0.04829358 0.1117206 0.05619587 0.007982489 0.02507312 0.03349619 0.01575294 0.02680348 0.02335575 0.02766681 0.03720479 0.02971207 0.03481135 0.03762075 0.1193895 0.04136778 0.3095488 0.03806171 0.05417495 0.04375112 0.144329 0.05144059 0.3861441 0.04491607 0.07003661 0.108209 0.1391186 0.1127831 0.2039894 0.1081036 0.1162216 0.05752893 0.06318651 0.05536051 0.07693626 0.05575184 0.05715865 0.007770032 0.06451125 0.01557529 0.1927278 0.008596145 0.03240892 0.01795947 0.1125866 0.03909418 0.2850895 0.0252202 0.06164596 0.03218241 0.05733104 0.03362629 0.093839 0.03321139 0.03915902 0.0104514 0.01466904 0.01372663 0.01391657 0.01163239 0.01500194 0.1131105 0.02455016 0.0460158 0.02646017 0.0348833 0.02885199 0.1332353 0.03858012 0.07850195 0.03430838 0.05728722 0.04660154 0.08544275 0.0276601 0.05140531 0.02480396 0.03822344 0.03249074 0.03223249 0.0259167 0.02671771 0.03235385 0.02639257 0.02639288 0.3444908 0.01230008 0.1272758 0.009164727 0.04199971 0.02279117 0.5127114 0.03037423 0.1867102 0.02430245 0.07916347 0.03931825 0.1920252 0.0198713 0.0677061 0.01349802 0.03950466 0.02317321 0.01660027 0.01341008 0.0141711 0.01223486 0.01307669 0.01368611 0.01649349 0.1155325 0.04126031 0.3112995 0.02441337 0.06608493 0.02081569 0.08964531 0.03112128 0.2392698 0.02435778 0.04809415 0.01513483 0.03907455 0.02276158 0.06818192 0.01779082 0.03013038 0.02687435 0.106475 0.04148795 0.2733178 0.03138539 0.0619568 0.02656696 0.04882158 0.03646567 0.06611806 0.03025403 0.04038842 0.02554426 0.04239234 0.03074304 0.05844103 0.026586 0.03551517 0.7871181 0.03296569 0.277536 0.027661 0.08997499 0.04876403 0.3685498 0.0269305 0.1137736 0.01872412 0.06574094 0.0337582 0.1113503 0.02812207 0.06868927 0.03324323 0.04115023 0.03313617 0.5320436 0.04604977 0.1975533 0.03018513 0.1107034 0.0551565 0.1247469 0.0254364 0.06583479 0.02384029 0.03693237 0.03121329 0.078693 0.03411972 0.04497596 0.03318185 0.04042994 0.03628108 0.024487Fit the Bayesian linear model 09
init_guess <- rep(0, ncol(Xmat_09)+1)
laplace_quad_09 <- my_laplace(init_guess, lm_logpost, info_09)
laplace_quad_09$converge## [1] "YES"laplace_quad_09$log_evidence## [1] 1218.426Display the posterior mode and posterior standard deviations
cat("posterior mode: ", laplace_quad_09$mode, "\n\n")## posterior mode: -0.1657045 0.03532588 0.005869348 0.04562268 0.05738303 0.05023855 0.01902506 0.005425965 0.008165973 -0.00189388 0.02175495 0.003596849 0.0007556507 0.2396139 0.6454933 0.1285493 0.01294464 0.04906861 0.09574578 0.03968679 -0.01621884 0.01773994 -0.005136824 -0.01238986 0.007694969 0.02452262 -0.03187443 0.09941477 -0.04764033 0.01165765 0.03131431 -0.005566811 -3.124398cat("posterior standard deviations: ", sqrt(diag(laplace_quad_09$var_matrix)))## posterior standard deviations: 0.0171563 0.008240401 0.01884723 0.01582103 0.01961775 0.01232755 0.01765138 0.008073312 0.005831563 0.006728248 0.006187497 0.006421094 0.006156087 0.007259569 0.01044355 0.009565383 0.005668725 0.004200285 0.00772694 0.0058205 0.003243595 0.008423175 0.007202855 0.009852411 0.008447183 0.007011812 0.009478973 0.004242994 0.006585712 0.008049399 0.005039858 0.003228666 0.02448401Calculate the posterior model weight
cat("log_evidence model 09: ", laplace_quad_09$log_evidence, "\n\n")## log_evidence model 09: 1218.426cat("log_evidence model 08: ", laplace_quad_08$log_evidence, "\n\n")## log_evidence model 08: -InfModel 09 is the best one in Laplace Approximation approach, Residual standard error: 0.04481
varphi_09 <- laplace_quad_09$mode[length(laplace_quad_09$mode)]
cat("posterior UNCERTAINTY model 09: ", exp(varphi_09), "\n")## posterior UNCERTAINTY model 09: 0.04396338posterior standard deviations: 0.02448401
Since 0.04481 and 0.04396338 are close, and the standard deviations is quite large compared with posterior uncertainty, we could say that it is uncertain about the σ
##Part ii: Regression – iiC) Linear models Predictions
viz_grid <- expand.grid(R = 0,
G = seq(-3, 3, length.out=75),
B = 0,
Hue = seq(-2.5, 2.5, length.out=6),
#Lightness = unique(df_standard$Lightness),
Lightness = "dark",
#Saturation = unique(df_standard$Saturation),
Saturation = "gray",
KEEP.OUT.ATTRS = FALSE,
stringsAsFactors = FALSE) %>%
as.data.frame() %>% tibble::as_tibble()
viz_grid %>% glimpse()## Rows: 450
## Columns: 6
## $ R <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ G <dbl> -3.000000, -2.918919, -2.837838, -2.756757, -2.675676, -2.5…
## $ B <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ Hue <dbl> -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5,…
## $ Lightness <chr> "dark", "dark", "dark", "dark", "dark", "dark", "dark", "da…
## $ Saturation <chr> "gray", "gray", "gray", "gray", "gray", "gray", "gray", "gr…tidy_predict <- function(mod, xnew)
{
pred_df <- predict(mod, xnew, interval = "confidence") %>%
as.data.frame() %>% tibble::as_tibble() %>%
dplyr::select(pred = fit, ci_lwr = lwr, ci_upr = upr) %>%
bind_cols(predict(mod, xnew, interval = 'prediction') %>%
as.data.frame() %>% tibble::as_tibble() %>%
dplyr::select(pred_lwr = lwr, pred_upr = upr))
xnew %>% bind_cols(pred_df)
}#Use non-Bayesian models for the predictions Make predictions with each of the models
pred_lm_09 <- tidy_predict(re_load_mod09, viz_grid)
pred_lm_09 %>% glimpse()## Rows: 450
## Columns: 11
## $ R <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ G <dbl> -3.000000, -2.918919, -2.837838, -2.756757, -2.675676, -2.5…
## $ B <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ Hue <dbl> -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5,…
## $ Lightness <chr> "dark", "dark", "dark", "dark", "dark", "dark", "dark", "da…
## $ Saturation <chr> "gray", "gray", "gray", "gray", "gray", "gray", "gray", "gr…
## $ pred <dbl> -1.1848697, -1.1834536, -1.1807786, -1.1768445, -1.1716515,…
## $ ci_lwr <dbl> -1.3401496, -1.3317272, -1.3222454, -1.3117049, -1.3001065,…
## $ ci_upr <dbl> -1.0295898, -1.0351801, -1.0393118, -1.0419842, -1.0431966,…
## $ pred_lwr <dbl> -1.3633269, -1.3558489, -1.3473557, -1.3378489, -1.3273300,…
## $ pred_upr <dbl> -1.0064126, -1.0110583, -1.0142014, -1.0158402, -1.0159731,…visualization for each model’s prediction
pred_lm_09 %>% ggplot(aes(x = G)) + geom_ribbon(aes(ymin = pred_lwr, ymax = pred_upr), fill = 'blue') + geom_ribbon(aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
geom_line(aes(y = pred)) + facet_wrap(~Hue) +
labs(x = "pred_lm_09$x1")
pred_lm_08 <- tidy_predict(re_load_mod08, viz_grid)compare the visualizations across models include a coord_cartesian() layer with the ylim argument set to c(-2,4)
pred_lm_08 %>% ggplot(aes(x = G)) + geom_ribbon(aes(ymin = pred_lwr, ymax = pred_upr), fill = 'blue') + geom_ribbon(aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
geom_line(aes(y = pred)) + facet_wrap(~Hue) + coord_cartesian(ylim = c(-2,4)) +
labs(x = "pred_lm_08$x1")
When R < 2 ,the predictive trends are similar between the 2 selected linear models.When R > 2, the predictive trends look different. The uncertainty of model 8 is big, showing that model 8 is over-fitting.
##Part ii: Regression – iiD) Train/tune with resampling • Linear models: • All categorical and continuous inputs - linear additive features • Add categorical inputs to all main effect and all pairwise interactions of continuous inputs • The 2 models selected from iiA) (if they are not one of the two above) • Regularized regression with Elastic net • Add categorical inputs to all main effect and all pairwise interactions of continuous inputs • The more complex of the 2 models selected from iiA) • Neural network • Random forest • Gradient boosted tree • 2 methods of your choice that we did not explicitly discuss in lecture You must use ALL categorical and continuous inputs with the non-linear methods
#specify the resampling scheme and primary performance metric
my_ctrl <- trainControl( method = "repeatedcv", number = 10, repeats = 3)
my_metric <- 'RMSE'You must train, assess, tune, and compare more complex methods
train_lm_01 <- train(y ~ .,
data = dfii,
method = "lm",
metric = my_metric,
preProcess = c("center", "scale"),
trControl = my_ctrl)train_lm_02 <- train(y ~ Lightness + Saturation + (R + G + B + Hue)^2,
data = dfii,
method = "lm",
metric = my_metric,
preProcess = c("center", "scale"),
trControl = my_ctrl)train_lm_09 <- train(y ~ Lightness + Saturation + R*G*B*Hue + I(R^2) + I(G^2) + I(B^2) + I(Hue^2),
data = dfii,
method = "lm",
metric = my_metric,
preProcess = c("center", "scale"),
trControl = my_ctrl)train_lm_09$results## intercept RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 TRUE 0.054563 0.9979538 0.04092225 0.006605376 0.0004091835 0.003634049train_lm_09 %>% coefplot()
train_lm_08 <- train(y ~ (Lightness + Saturation) * (( R + G + B + Hue)^2 + I(R^2) + I(G^2) + I(B^2) + I(Hue^2)),
data = dfii,
method = "lm",
metric = my_metric,
preProcess = c("center", "scale"),
trControl = my_ctrl)train_lm_08$results## intercept RMSE Rsquared MAE RMSESD RsquaredSD
## 1 TRUE 0.04699707 0.9984425 0.03373372 0.007872184 0.0004992378
## MAESD
## 1 0.003943451#Regularized regression with Elastic net Train, assess, and tune the glmnet elastic net model with the defined resampling scheme. Assign the result to the enet_default object and display the result to the screen.
set.seed(1234)
enet_default_01 <- caret::train(y ~ Lightness + Saturation + (R + G + B + Hue)^2,
data = dfii,
method = 'glmnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl)Create a custom tuning grid to further tune the elastic net lambda and alpha tuning parameters.
lambda_seq <- exp(seq(log(min(enet_default_01$results$lambda)),
log(max(enet_default_01$results$lambda)),
length.out = 25))
enet_grid_01 <- expand.grid(alpha = seq(0.0, 0.15, by = .01), lambda = lambda_seq)Train, assess, and tune the elastic net model with the custom tuning grid and assign the result to the enet_tune_01 object.
set.seed(1234)
enet_tune_01 <- caret::train(y ~ Lightness + Saturation + (R + G + B + Hue)^2,
data = dfii,
method = 'glmnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl,
tuneGrid = enet_grid_01)
enet_tune_01$bestTune## alpha lambda
## 78 0.03 0.003412554model 09
set.seed(1234)
enet_default_09 <- caret::train(y ~ Lightness + Saturation + R * G * B * Hue + I(R^2) + I(G^2) + I(B^2) + I(Hue^2),
data = dfii,
method = 'glmnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl)
enet_default_09$bestTune## alpha lambda
## 1 0.1 0.002331673lambda_seq <- exp(seq(log(min(enet_default_09$results$lambda)),
log(max(enet_default_09$results$lambda)),
length.out = 50))
enet_grid_09 <- expand.grid(alpha = seq(0.03, 0.12, by = .01), lambda = lambda_seq)
set.seed(1234)
enet_tune_09 <- caret::train(y ~ Lightness + Saturation + R * G * B * Hue + I(R^2) + I(G^2) + I(B^2) + I(Hue^2),
data = dfii,
method = 'glmnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl,
tuneGrid = enet_grid_09)
enet_tune_09$bestTune## alpha lambda
## 53 0.04 0.002813845model 08
set.seed(1234)
enet_default_08 <- caret::train(y ~ (Lightness + Saturation) * (( R + G + B + Hue)^2 + I(R^2) + I(G^2) + I(B^2) + I(Hue^2)),
data = dfii,
method = 'glmnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl)
enet_default_08$bestTune## alpha lambda
## 1 0.1 0.002331673lambda_seq <- exp(seq(log(min(enet_default_08$results$lambda)),
log(max(enet_default_08$results$lambda)),
length.out = 50))
enet_grid_08 <- expand.grid(alpha = seq(0.03, 0.12, by = .01), lambda = lambda_seq)
set.seed(1234)
enet_tune_08 <- caret::train(y ~ (Lightness + Saturation) * (( R + G + B + Hue)^2 + I(R^2) + I(G^2) + I(B^2) + I(Hue^2)),
data = dfii,
method = 'glmnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl,
tuneGrid = enet_grid_08)
enet_tune_08$bestTune## alpha lambda
## 53 0.04 0.002813845enet_tune_08$results## alpha lambda RMSE Rsquared MAE RMSESD RsquaredSD
## 1 0.03 0.002331673 0.06469133 0.9970976 0.04831645 0.008261880 0.0007151921
## 2 0.03 0.002561438 0.06469133 0.9970976 0.04831645 0.008261880 0.0007151921
## 3 0.03 0.002813845 0.06469133 0.9970976 0.04831645 0.008261880 0.0007151921
## 4 0.03 0.003091125 0.06469133 0.9970976 0.04831645 0.008261880 0.0007151921
## 5 0.03 0.003395728 0.06469133 0.9970976 0.04831645 0.008261880 0.0007151921
## 6 0.03 0.003730347 0.06469133 0.9970976 0.04831645 0.008261880 0.0007151921
## 7 0.03 0.004097939 0.06475364 0.9970925 0.04837121 0.008239364 0.0007131218
## 8 0.03 0.004501755 0.06488398 0.9970815 0.04848669 0.008202778 0.0007098957
## 9 0.03 0.004945363 0.06505613 0.9970668 0.04864269 0.008154320 0.0007057924
## 10 0.03 0.005432685 0.06526198 0.9970494 0.04883199 0.008104553 0.0007009719
## 11 0.03 0.005968028 0.06550002 0.9970292 0.04904412 0.008054690 0.0006955343
## 12 0.03 0.006556124 0.06576161 0.9970069 0.04926500 0.008004048 0.0006898887
## 13 0.03 0.007202172 0.06604178 0.9969829 0.04949372 0.007961838 0.0006848487
## 14 0.03 0.007911882 0.06633855 0.9969571 0.04973025 0.007937011 0.0006814158
## 15 0.03 0.008691528 0.06668191 0.9969264 0.05000907 0.007961344 0.0006835732
## 16 0.03 0.009548001 0.06732573 0.9968678 0.05053250 0.008032438 0.0006923369
## 17 0.03 0.010488872 0.06800293 0.9968067 0.05108916 0.008046858 0.0006951498
## 18 0.03 0.011522457 0.06870501 0.9967414 0.05167610 0.008114937 0.0007030406
## 19 0.03 0.012657894 0.06955303 0.9966615 0.05238087 0.008243988 0.0007174494
## 20 0.03 0.013905217 0.07037429 0.9965833 0.05305938 0.008326494 0.0007278607
## 21 0.03 0.015275453 0.07107525 0.9965162 0.05361570 0.008397099 0.0007360847
## 22 0.03 0.016780713 0.07173281 0.9964529 0.05415292 0.008487554 0.0007449029
## 23 0.03 0.018434304 0.07237017 0.9963911 0.05467640 0.008575698 0.0007529826
## 24 0.03 0.020250841 0.07298426 0.9963314 0.05517455 0.008648617 0.0007597649
## 25 0.03 0.022246382 0.07359752 0.9962713 0.05565850 0.008756934 0.0007699943
## 26 0.03 0.024438566 0.07423341 0.9962088 0.05614753 0.008862872 0.0007795367
## 27 0.03 0.026846770 0.07492348 0.9961405 0.05665576 0.008970189 0.0007898898
## 28 0.03 0.029492282 0.07564245 0.9960688 0.05717438 0.009081163 0.0008011827
## 29 0.03 0.032398485 0.07637916 0.9959949 0.05767260 0.009197069 0.0008127764
## 30 0.03 0.035591069 0.07715558 0.9959171 0.05816467 0.009312992 0.0008243812
## 31 0.03 0.039098253 0.07798333 0.9958341 0.05866270 0.009449550 0.0008368006
## 32 0.03 0.042951040 0.07887029 0.9957448 0.05917442 0.009603052 0.0008511588
## 33 0.03 0.047183484 0.07983058 0.9956480 0.05970632 0.009730927 0.0008645438
## 34 0.03 0.051832999 0.08087562 0.9955425 0.06025165 0.009888193 0.0008793951
## 35 0.03 0.056940682 0.08199837 0.9954283 0.06082387 0.010056117 0.0008967142
## 36 0.03 0.062551681 0.08317393 0.9953086 0.06140897 0.010255160 0.0009182984
## 37 0.03 0.068715596 0.08454000 0.9951672 0.06211804 0.010496599 0.0009450340
## 38 0.03 0.075486909 0.08609745 0.9950036 0.06294213 0.010730461 0.0009737421
## 39 0.03 0.082925475 0.08776279 0.9948269 0.06382311 0.010999936 0.0010062546
## 40 0.03 0.091097046 0.08954004 0.9946373 0.06478202 0.011242468 0.0010382843
## 41 0.03 0.100073853 0.09146241 0.9944310 0.06583223 0.011525035 0.0010759047
## 42 0.03 0.109935245 0.09357598 0.9942013 0.06705153 0.011831727 0.0011171873
## 43 0.03 0.120768389 0.09591907 0.9939421 0.06844893 0.012133769 0.0011633467
## 44 0.03 0.132669044 0.09837418 0.9936692 0.06995364 0.012453260 0.0012111924
## 45 0.03 0.145742403 0.10100331 0.9933730 0.07158695 0.012791449 0.0012629898
## 46 0.03 0.160104026 0.10382156 0.9930526 0.07339517 0.013121470 0.0013170598
## 47 0.03 0.175880860 0.10682344 0.9927072 0.07539176 0.013452182 0.0013754914
## 48 0.03 0.193212361 0.11000161 0.9923388 0.07760260 0.013800554 0.0014358709
## 49 0.03 0.212251727 0.11341051 0.9919403 0.08002810 0.014161212 0.0015000165
## 50 0.03 0.233167255 0.11702212 0.9915150 0.08260372 0.014551578 0.0015711800
## 51 0.04 0.002331673 0.06434578 0.9971271 0.04827285 0.008482254 0.0007185286
## 52 0.04 0.002561438 0.06434578 0.9971271 0.04827285 0.008482254 0.0007185286
## 53 0.04 0.002813845 0.06434578 0.9971271 0.04827285 0.008482254 0.0007185286
## 54 0.04 0.003091125 0.06441981 0.9971211 0.04832502 0.008423029 0.0007142749
## 55 0.04 0.003395728 0.06455125 0.9971105 0.04842608 0.008321999 0.0007069920
## 56 0.04 0.003730347 0.06468372 0.9970998 0.04852912 0.008230164 0.0006999996
## 57 0.04 0.004097939 0.06484274 0.9970868 0.04864502 0.008147612 0.0006932029
## 58 0.04 0.004501755 0.06506585 0.9970682 0.04880313 0.008076719 0.0006869343
## 59 0.04 0.004945363 0.06532835 0.9970459 0.04899824 0.008014006 0.0006813712
## 60 0.04 0.005432685 0.06562305 0.9970205 0.04922407 0.007957927 0.0006768883
## 61 0.04 0.005968028 0.06593262 0.9969936 0.04946291 0.007919397 0.0006740885
## 62 0.04 0.006556124 0.06627152 0.9969634 0.04972765 0.007914773 0.0006757677
## 63 0.04 0.007202172 0.06666023 0.9969278 0.05003858 0.007967327 0.0006844705
## 64 0.04 0.007911882 0.06714814 0.9968834 0.05042398 0.008012910 0.0006911499
## 65 0.04 0.008691528 0.06783967 0.9968204 0.05099654 0.008038191 0.0006966698
## 66 0.04 0.009548001 0.06861943 0.9967475 0.05167032 0.008106574 0.0007067219
## 67 0.04 0.010488872 0.06935469 0.9966783 0.05230262 0.008189705 0.0007157470
## 68 0.04 0.011522457 0.07012753 0.9966054 0.05297301 0.008245447 0.0007223625
## 69 0.04 0.012657894 0.07075826 0.9965450 0.05350044 0.008322799 0.0007301280
## 70 0.04 0.013905217 0.07133317 0.9964897 0.05396697 0.008402139 0.0007375859
## 71 0.04 0.015275453 0.07190271 0.9964346 0.05440702 0.008502945 0.0007455987
## 72 0.04 0.016780713 0.07246828 0.9963794 0.05485642 0.008592201 0.0007540615
## 73 0.04 0.018434304 0.07305463 0.9963219 0.05532690 0.008670970 0.0007610559
## 74 0.04 0.020250841 0.07366406 0.9962617 0.05581267 0.008761015 0.0007696828
## 75 0.04 0.022246382 0.07432874 0.9961956 0.05632453 0.008873265 0.0007798014
## 76 0.04 0.024438566 0.07501312 0.9961273 0.05683575 0.008978684 0.0007896267
## 77 0.04 0.026846770 0.07573066 0.9960559 0.05734978 0.009062715 0.0007975186
## 78 0.04 0.029492282 0.07647109 0.9959815 0.05784901 0.009176572 0.0008091245
## 79 0.04 0.032398485 0.07722404 0.9959056 0.05832678 0.009298081 0.0008209874
## 80 0.04 0.035591069 0.07803870 0.9958234 0.05882226 0.009423555 0.0008323850
## 81 0.04 0.039098253 0.07895688 0.9957298 0.05936800 0.009564534 0.0008461421
## 82 0.04 0.042951040 0.07998569 0.9956239 0.05996086 0.009715191 0.0008610001
## 83 0.04 0.047183484 0.08101113 0.9955192 0.06049000 0.009872476 0.0008773812
## 84 0.04 0.051832999 0.08208157 0.9954097 0.06101179 0.010103233 0.0009011023
## 85 0.04 0.056940682 0.08320820 0.9952954 0.06152979 0.010300118 0.0009222127
## 86 0.04 0.062551681 0.08442055 0.9951726 0.06210111 0.010491548 0.0009401433
## 87 0.04 0.068715596 0.08583528 0.9950259 0.06282401 0.010753382 0.0009677290
## 88 0.04 0.075486909 0.08739317 0.9948632 0.06364379 0.010965116 0.0009900960
## 89 0.04 0.082925475 0.08903411 0.9946904 0.06451960 0.011200444 0.0010185264
## 90 0.04 0.091097046 0.09084925 0.9944988 0.06551424 0.011485257 0.0010519001
## 91 0.04 0.100073853 0.09278169 0.9942949 0.06659743 0.011715324 0.0010810017
## 92 0.04 0.109935245 0.09476850 0.9940848 0.06776117 0.012020034 0.0011163142
## 93 0.04 0.120768389 0.09697151 0.9938495 0.06911267 0.012311410 0.0011532262
## 94 0.04 0.132669044 0.09932890 0.9935954 0.07060709 0.012618902 0.0011945841
## 95 0.04 0.145742403 0.10186992 0.9933197 0.07226225 0.012968620 0.0012419858
## 96 0.04 0.160104026 0.10464836 0.9930151 0.07409249 0.013341021 0.0012969680
## 97 0.04 0.175880860 0.10768922 0.9926777 0.07614422 0.013732890 0.0013582646
## 98 0.04 0.193212361 0.11094358 0.9923138 0.07841133 0.014106815 0.0014231530
## 99 0.04 0.212251727 0.11444213 0.9919178 0.08093087 0.014501349 0.0014937678
## 100 0.04 0.233167255 0.11823264 0.9914837 0.08367532 0.014917187 0.0015720254
## 101 0.05 0.002331673 0.06599745 0.9969818 0.04963519 0.008091300 0.0007098718
## 102 0.05 0.002561438 0.06602428 0.9969798 0.04965176 0.008064375 0.0007077651
## 103 0.05 0.002813845 0.06606825 0.9969761 0.04968186 0.008045238 0.0007065042
## 104 0.05 0.003091125 0.06612997 0.9969709 0.04972540 0.008023859 0.0007044474
## 105 0.05 0.003395728 0.06619682 0.9969653 0.04977131 0.008003177 0.0007022803
## 106 0.05 0.003730347 0.06626809 0.9969593 0.04982267 0.007982522 0.0006998703
## 107 0.05 0.004097939 0.06634565 0.9969528 0.04987700 0.007965617 0.0006977876
## 108 0.05 0.004501755 0.06642619 0.9969459 0.04993325 0.007955470 0.0006960427
## 109 0.05 0.004945363 0.06651275 0.9969384 0.04999205 0.007947710 0.0006944166
## 110 0.05 0.005432685 0.06662503 0.9969287 0.05006444 0.007926683 0.0006915255
## 111 0.05 0.005968028 0.06675902 0.9969171 0.05015266 0.007895290 0.0006881072
## 112 0.05 0.006556124 0.06697096 0.9968984 0.05030813 0.007906626 0.0006876593
## 113 0.05 0.007202172 0.06753036 0.9968473 0.05077383 0.008086236 0.0007012695
## 114 0.05 0.007911882 0.06823725 0.9967822 0.05135319 0.008098162 0.0007025520
## 115 0.05 0.008691528 0.06896331 0.9967149 0.05202077 0.008124520 0.0007073545
## 116 0.05 0.009548001 0.06964351 0.9966501 0.05263200 0.008200716 0.0007168821
## 117 0.05 0.010488872 0.07026779 0.9965911 0.05318459 0.008223317 0.0007203124
## 118 0.05 0.011522457 0.07081931 0.9965381 0.05364650 0.008335262 0.0007294368
## 119 0.05 0.012657894 0.07135380 0.9964865 0.05406004 0.008422962 0.0007364609
## 120 0.05 0.013905217 0.07185256 0.9964380 0.05444180 0.008504849 0.0007435492
## 121 0.05 0.015275453 0.07239734 0.9963847 0.05485109 0.008611972 0.0007536311
## 122 0.05 0.016780713 0.07299284 0.9963261 0.05532754 0.008695805 0.0007603695
## 123 0.05 0.018434304 0.07364642 0.9962615 0.05585836 0.008799177 0.0007693243
## 124 0.05 0.020250841 0.07430506 0.9961960 0.05636927 0.008902381 0.0007795311
## 125 0.05 0.022246382 0.07498425 0.9961283 0.05687850 0.008980316 0.0007872723
## 126 0.05 0.024438566 0.07566466 0.9960606 0.05735333 0.009069558 0.0007952494
## 127 0.05 0.026846770 0.07630035 0.9959972 0.05775526 0.009164997 0.0008044931
## 128 0.05 0.029492282 0.07700627 0.9959266 0.05819048 0.009284266 0.0008154791
## 129 0.05 0.032398485 0.07781705 0.9958446 0.05866912 0.009431321 0.0008291074
## 130 0.05 0.035591069 0.07871840 0.9957527 0.05919032 0.009591654 0.0008451463
## 131 0.05 0.039098253 0.07972159 0.9956497 0.05975180 0.009769208 0.0008636660
## 132 0.05 0.042951040 0.08074327 0.9955451 0.06027815 0.009951459 0.0008821962
## 133 0.05 0.047183484 0.08176965 0.9954407 0.06076243 0.010141032 0.0009003061
## 134 0.05 0.051832999 0.08285937 0.9953301 0.06126446 0.010304600 0.0009159240
## 135 0.05 0.056940682 0.08407021 0.9952066 0.06184138 0.010457395 0.0009299067
## 136 0.05 0.062551681 0.08538090 0.9950721 0.06248948 0.010671923 0.0009518402
## 137 0.05 0.068715596 0.08679432 0.9949266 0.06322470 0.010907073 0.0009764365
## 138 0.05 0.075486909 0.08830705 0.9947706 0.06402878 0.011136104 0.0009998740
## 139 0.05 0.082925475 0.08994124 0.9946005 0.06490908 0.011380280 0.0010265879
## 140 0.05 0.091097046 0.09171479 0.9944153 0.06588035 0.011631348 0.0010555143
## 141 0.05 0.100073853 0.09362466 0.9942153 0.06694782 0.011905996 0.0010878648
## 142 0.05 0.109935245 0.09566984 0.9940013 0.06816869 0.012221274 0.0011242868
## 143 0.05 0.120768389 0.09792021 0.9937634 0.06957544 0.012584526 0.0011690856
## 144 0.05 0.132669044 0.10042109 0.9934948 0.07118747 0.012939611 0.0012174890
## 145 0.05 0.145742403 0.10312171 0.9932019 0.07295712 0.013299653 0.0012699011
## 146 0.05 0.160104026 0.10601626 0.9928860 0.07490088 0.013652811 0.0013246111
## 147 0.05 0.175880860 0.10909823 0.9925479 0.07705657 0.014035496 0.0013853568
## 148 0.05 0.193212361 0.11241231 0.9921815 0.07941359 0.014455983 0.0014558492
## 149 0.05 0.212251727 0.11600648 0.9917814 0.08198735 0.014905317 0.0015346012
## 150 0.05 0.233167255 0.11991603 0.9913420 0.08479580 0.015368550 0.0016200758
## 151 0.06 0.002331673 0.06602278 0.9969886 0.04941966 0.007563829 0.0006431239
## 152 0.06 0.002561438 0.06611745 0.9969797 0.04950308 0.007570636 0.0006450830
## 153 0.06 0.002813845 0.06623065 0.9969693 0.04960837 0.007571432 0.0006463919
## 154 0.06 0.003091125 0.06632374 0.9969605 0.04969565 0.007590679 0.0006500107
## 155 0.06 0.003395728 0.06644922 0.9969487 0.04981232 0.007607122 0.0006531666
## 156 0.06 0.003730347 0.06656775 0.9969376 0.04992154 0.007631107 0.0006570678
## 157 0.06 0.004097939 0.06668564 0.9969265 0.05002891 0.007657050 0.0006612038
## 158 0.06 0.004501755 0.06679490 0.9969161 0.05013256 0.007697403 0.0006669382
## 159 0.06 0.004945363 0.06695141 0.9969014 0.05028095 0.007750499 0.0006733131
## 160 0.06 0.005432685 0.06714537 0.9968834 0.05044851 0.007896301 0.0006854559
## 161 0.06 0.005968028 0.06730687 0.9968680 0.05060760 0.007990098 0.0006955940
## 162 0.06 0.006556124 0.06764251 0.9968365 0.05089391 0.008119790 0.0007071140
## 163 0.06 0.007202172 0.06845384 0.9967618 0.05160845 0.008181530 0.0007118695
## 164 0.06 0.007911882 0.06902702 0.9967090 0.05212574 0.008129115 0.0007103231
## 165 0.06 0.008691528 0.06967411 0.9966471 0.05269914 0.008257297 0.0007212138
## 166 0.06 0.009548001 0.07023437 0.9965941 0.05319590 0.008295085 0.0007254485
## 167 0.06 0.010488872 0.07078441 0.9965412 0.05364472 0.008377449 0.0007321204
## 168 0.06 0.011522457 0.07129681 0.9964913 0.05404522 0.008482651 0.0007408705
## 169 0.06 0.012657894 0.07182049 0.9964399 0.05444373 0.008584286 0.0007501632
## 170 0.06 0.013905217 0.07241434 0.9963812 0.05490674 0.008678387 0.0007585683
## 171 0.06 0.015275453 0.07302105 0.9963215 0.05539766 0.008752197 0.0007646374
## 172 0.06 0.016780713 0.07362670 0.9962614 0.05587272 0.008842403 0.0007733790
## 173 0.06 0.018434304 0.07425578 0.9961986 0.05634394 0.008940841 0.0007831618
## 174 0.06 0.020250841 0.07484683 0.9961398 0.05676153 0.009000271 0.0007896382
## 175 0.06 0.022246382 0.07543054 0.9960819 0.05715132 0.009080600 0.0007969074
## 176 0.06 0.024438566 0.07602455 0.9960232 0.05752811 0.009197609 0.0008072066
## 177 0.06 0.026846770 0.07673839 0.9959513 0.05797472 0.009331806 0.0008203594
## 178 0.06 0.029492282 0.07758495 0.9958654 0.05849610 0.009466136 0.0008336953
## 179 0.06 0.032398485 0.07852664 0.9957692 0.05905735 0.009621772 0.0008490176
## 180 0.06 0.035591069 0.07954945 0.9956639 0.05964251 0.009783569 0.0008659694
## 181 0.06 0.039098253 0.08051941 0.9955652 0.06013775 0.009917051 0.0008783735
## 182 0.06 0.042951040 0.08149988 0.9954658 0.06058246 0.010073395 0.0008915929
## 183 0.06 0.047183484 0.08256780 0.9953569 0.06106842 0.010207269 0.0009041966
## 184 0.06 0.051832999 0.08366278 0.9952462 0.06158260 0.010378338 0.0009208422
## 185 0.06 0.056940682 0.08481650 0.9951303 0.06214566 0.010567508 0.0009360087
## 186 0.06 0.062551681 0.08608842 0.9950018 0.06278100 0.010800635 0.0009569520
## 187 0.06 0.068715596 0.08749228 0.9948588 0.06348639 0.011020333 0.0009790133
## 188 0.06 0.075486909 0.08899714 0.9947051 0.06428019 0.011246827 0.0010014661
## 189 0.06 0.082925475 0.09063926 0.9945360 0.06517749 0.011516199 0.0010308975
## 190 0.06 0.091097046 0.09246986 0.9943456 0.06623117 0.011826741 0.0010651070
## 191 0.06 0.100073853 0.09449794 0.9941328 0.06742253 0.012155910 0.0011037005
## 192 0.06 0.109935245 0.09670719 0.9938998 0.06875474 0.012494715 0.0011442902
## 193 0.06 0.120768389 0.09911346 0.9936435 0.07029468 0.012868094 0.0011917414
## 194 0.06 0.132669044 0.10174091 0.9933598 0.07201514 0.013238154 0.0012423063
## 195 0.06 0.145742403 0.10452543 0.9930577 0.07384551 0.013603040 0.0012974234
## 196 0.06 0.160104026 0.10747476 0.9927377 0.07588663 0.013983449 0.0013578085
## 197 0.06 0.175880860 0.11065581 0.9923923 0.07815043 0.014418624 0.0014265708
## 198 0.06 0.193212361 0.11411765 0.9920133 0.08056964 0.014871175 0.0015022157
## 199 0.06 0.212251727 0.11786257 0.9915997 0.08318840 0.015334734 0.0015853880
## 200 0.06 0.233167255 0.12189773 0.9911518 0.08605963 0.015797891 0.0016745397
## 201 0.07 0.002331673 0.06612137 0.9969781 0.04953104 0.007817640 0.0006656349
## 202 0.07 0.002561438 0.06616965 0.9969737 0.04957278 0.007803131 0.0006651802
## 203 0.07 0.002813845 0.06621951 0.9969691 0.04961458 0.007789929 0.0006648916
## 204 0.07 0.003091125 0.06627128 0.9969643 0.04965654 0.007777268 0.0006647276
## 205 0.07 0.003395728 0.06632642 0.9969592 0.04970156 0.007765852 0.0006646424
## 206 0.07 0.003730347 0.06641224 0.9969512 0.04978470 0.007773234 0.0006654805
## 207 0.07 0.004097939 0.06651188 0.9969417 0.04988280 0.007798032 0.0006684866
## 208 0.07 0.004501755 0.06666180 0.9969277 0.05000662 0.007873719 0.0006752304
## 209 0.07 0.004945363 0.06689524 0.9969044 0.05023632 0.008035583 0.0006950039
## 210 0.07 0.005432685 0.06724710 0.9968712 0.05057148 0.008160660 0.0007088048
## 211 0.07 0.005968028 0.06778992 0.9968226 0.05104959 0.008127031 0.0007098763
## 212 0.07 0.006556124 0.06843141 0.9967644 0.05163841 0.008143265 0.0007082619
## 213 0.07 0.007202172 0.06905435 0.9967057 0.05220010 0.008162190 0.0007132757
## 214 0.07 0.007911882 0.06967379 0.9966471 0.05274207 0.008240185 0.0007206087
## 215 0.07 0.008691528 0.07021778 0.9965951 0.05320744 0.008317084 0.0007270094
## 216 0.07 0.009548001 0.07076055 0.9965425 0.05364820 0.008412619 0.0007349014
## 217 0.07 0.010488872 0.07125137 0.9964947 0.05402042 0.008489388 0.0007423414
## 218 0.07 0.011522457 0.07181447 0.9964394 0.05444734 0.008598158 0.0007521709
## 219 0.07 0.012657894 0.07238322 0.9963834 0.05489409 0.008656014 0.0007573683
## 220 0.07 0.013905217 0.07295714 0.9963265 0.05535403 0.008722185 0.0007634652
## 221 0.07 0.015275453 0.07354572 0.9962676 0.05580146 0.008814622 0.0007721347
## 222 0.07 0.016780713 0.07412682 0.9962097 0.05621426 0.008885228 0.0007788746
## 223 0.07 0.018434304 0.07469651 0.9961530 0.05660842 0.008959421 0.0007861081
## 224 0.07 0.020250841 0.07525537 0.9960977 0.05698338 0.009057574 0.0007956502
## 225 0.07 0.022246382 0.07585182 0.9960387 0.05736943 0.009169251 0.0008066654
## 226 0.07 0.024438566 0.07656039 0.9959678 0.05782690 0.009304066 0.0008185226
## 227 0.07 0.026846770 0.07740513 0.9958821 0.05836042 0.009443917 0.0008317912
## 228 0.07 0.029492282 0.07840265 0.9957790 0.05897594 0.009618125 0.0008491306
## 229 0.07 0.032398485 0.07942030 0.9956742 0.05956770 0.009720600 0.0008604750
## 230 0.07 0.035591069 0.08031375 0.9955837 0.06000058 0.009830362 0.0008697570
## 231 0.07 0.039098253 0.08124529 0.9954889 0.06041402 0.009942743 0.0008798949
## 232 0.07 0.042951040 0.08221867 0.9953905 0.06087064 0.010105133 0.0008929247
## 233 0.07 0.047183484 0.08322177 0.9952898 0.06134719 0.010267893 0.0009062638
## 234 0.07 0.051832999 0.08427383 0.9951848 0.06183901 0.010482143 0.0009238223
## 235 0.07 0.056940682 0.08543029 0.9950702 0.06238168 0.010676276 0.0009397898
## 236 0.07 0.062551681 0.08672574 0.9949410 0.06302533 0.010869751 0.0009564904
## 237 0.07 0.068715596 0.08810992 0.9948017 0.06373288 0.011132968 0.0009813097
## 238 0.07 0.075486909 0.08964941 0.9946452 0.06457386 0.011410884 0.0010077414
## 239 0.07 0.082925475 0.09140004 0.9944636 0.06558534 0.011727221 0.0010416101
## 240 0.07 0.091097046 0.09336444 0.9942577 0.06673917 0.012040668 0.0010773610
## 241 0.07 0.100073853 0.09552366 0.9940290 0.06802582 0.012396760 0.0011198009
## 242 0.07 0.109935245 0.09786501 0.9937800 0.06946474 0.012766315 0.0011643068
## 243 0.07 0.120768389 0.10036740 0.9935118 0.07107692 0.013173796 0.0012193576
## 244 0.07 0.132669044 0.10308972 0.9932171 0.07286831 0.013538150 0.0012721559
## 245 0.07 0.145742403 0.10592823 0.9929121 0.07479546 0.013919656 0.0013293807
## 246 0.07 0.160104026 0.10898381 0.9925838 0.07692166 0.014386505 0.0013992072
## 247 0.07 0.175880860 0.11233348 0.9922192 0.07927386 0.014837174 0.0014743433
## 248 0.07 0.193212361 0.11597074 0.9918209 0.08180541 0.015302054 0.0015570969
## 249 0.07 0.212251727 0.11988129 0.9913917 0.08452779 0.015752251 0.0016430901
## 250 0.07 0.233167255 0.12408848 0.9909292 0.08753082 0.016171684 0.0017320212
## 251 0.08 0.002331673 0.06540228 0.9970398 0.04905608 0.008159286 0.0007038839
## 252 0.08 0.002561438 0.06552516 0.9970292 0.04915486 0.008120710 0.0007001275
## 253 0.08 0.002813845 0.06568983 0.9970150 0.04928526 0.008075543 0.0006958692
## 254 0.08 0.003091125 0.06585061 0.9970009 0.04940661 0.008043689 0.0006928184
## 255 0.08 0.003395728 0.06602215 0.9969856 0.04953473 0.008026308 0.0006909039
## 256 0.08 0.003730347 0.06622089 0.9969680 0.04969136 0.007995982 0.0006879498
## 257 0.08 0.004097939 0.06646926 0.9969462 0.04990221 0.007964757 0.0006849660
## 258 0.08 0.004501755 0.06675436 0.9969202 0.05016849 0.007959621 0.0006857318
## 259 0.08 0.004945363 0.06717827 0.9968793 0.05053837 0.008076469 0.0007035679
## 260 0.08 0.005432685 0.06775667 0.9968256 0.05102232 0.008135539 0.0007109782
## 261 0.08 0.005968028 0.06847941 0.9967605 0.05171246 0.008066147 0.0007036704
## 262 0.08 0.006556124 0.06901161 0.9967098 0.05219179 0.008171773 0.0007168615
## 263 0.08 0.007202172 0.06958915 0.9966556 0.05268474 0.008249955 0.0007212972
## 264 0.08 0.007911882 0.07011654 0.9966054 0.05314027 0.008312999 0.0007260018
## 265 0.08 0.008691528 0.07064633 0.9965537 0.05357187 0.008420835 0.0007353479
## 266 0.08 0.009548001 0.07114233 0.9965053 0.05395102 0.008501990 0.0007424173
## 267 0.08 0.010488872 0.07169173 0.9964513 0.05437073 0.008592301 0.0007504858
## 268 0.08 0.011522457 0.07222679 0.9963986 0.05477859 0.008670318 0.0007578489
## 269 0.08 0.012657894 0.07278954 0.9963424 0.05521174 0.008735262 0.0007643609
## 270 0.08 0.013905217 0.07337058 0.9962843 0.05564600 0.008806676 0.0007716484
## 271 0.08 0.015275453 0.07396058 0.9962250 0.05608067 0.008896602 0.0007808646
## 272 0.08 0.016780713 0.07453183 0.9961676 0.05648016 0.008980839 0.0007898439
## 273 0.08 0.018434304 0.07510653 0.9961106 0.05687358 0.009055951 0.0007971203
## 274 0.08 0.020250841 0.07571244 0.9960507 0.05727809 0.009155105 0.0008067503
## 275 0.08 0.022246382 0.07638252 0.9959842 0.05770316 0.009279810 0.0008179515
## 276 0.08 0.024438566 0.07722920 0.9958982 0.05824720 0.009426767 0.0008304094
## 277 0.08 0.026846770 0.07820104 0.9957980 0.05886034 0.009553771 0.0008434462
## 278 0.08 0.029492282 0.07917899 0.9956969 0.05942298 0.009652648 0.0008543453
## 279 0.08 0.032398485 0.08001575 0.9956122 0.05982200 0.009745541 0.0008616317
## 280 0.08 0.035591069 0.08087638 0.9955253 0.06021482 0.009870375 0.0008719495
## 281 0.08 0.039098253 0.08175600 0.9954372 0.06063088 0.010021810 0.0008835896
## 282 0.08 0.042951040 0.08269281 0.9953437 0.06105697 0.010187084 0.0008961977
## 283 0.08 0.047183484 0.08371247 0.9952419 0.06151452 0.010367445 0.0009104129
## 284 0.08 0.051832999 0.08480353 0.9951337 0.06200826 0.010545794 0.0009242729
## 285 0.08 0.056940682 0.08597817 0.9950175 0.06258245 0.010748156 0.0009403672
## 286 0.08 0.062551681 0.08727611 0.9948886 0.06326164 0.011009019 0.0009623845
## 287 0.08 0.068715596 0.08872837 0.9947421 0.06404774 0.011296236 0.0009888688
## 288 0.08 0.075486909 0.09037166 0.9945732 0.06496592 0.011591625 0.0010187218
## 289 0.08 0.082925475 0.09223233 0.9943793 0.06603872 0.011919396 0.0010546124
## 290 0.08 0.091097046 0.09431118 0.9941596 0.06726403 0.012280611 0.0010963403
## 291 0.08 0.100073853 0.09656979 0.9939190 0.06862568 0.012643001 0.0011407982
## 292 0.08 0.109935245 0.09900091 0.9936592 0.07014717 0.013031878 0.0011905348
## 293 0.08 0.120768389 0.10160167 0.9933806 0.07182908 0.013440095 0.0012441732
## 294 0.08 0.132669044 0.10437139 0.9930837 0.07370031 0.013855710 0.0013023187
## 295 0.08 0.145742403 0.10734614 0.9927645 0.07577738 0.014296918 0.0013683976
## 296 0.08 0.160104026 0.11058474 0.9924140 0.07802877 0.014762359 0.0014426608
## 297 0.08 0.175880860 0.11408562 0.9920335 0.08043973 0.015223017 0.0015219700
## 298 0.08 0.193212361 0.11781969 0.9916285 0.08301066 0.015652759 0.0016027948
## 299 0.08 0.212251727 0.12165344 0.9912224 0.08572912 0.015956923 0.0016705477
## 300 0.08 0.233167255 0.12545817 0.9908456 0.08850224 0.016223528 0.0017290777
## 301 0.09 0.002331673 0.06450885 0.9971127 0.04837535 0.008196725 0.0007038527
## 302 0.09 0.002561438 0.06475829 0.9970917 0.04858027 0.008160155 0.0007022357
## 303 0.09 0.002813845 0.06501201 0.9970710 0.04878145 0.008074917 0.0006964606
## 304 0.09 0.003091125 0.06527306 0.9970491 0.04898692 0.008006119 0.0006926078
## 305 0.09 0.003395728 0.06560081 0.9970204 0.04924579 0.007984734 0.0006935216
## 306 0.09 0.003730347 0.06600725 0.9969846 0.04957253 0.007968207 0.0006939355
## 307 0.09 0.004097939 0.06647671 0.9969428 0.04995438 0.007955978 0.0006944053
## 308 0.09 0.004501755 0.06702724 0.9968931 0.05043546 0.007996410 0.0006985259
## 309 0.09 0.004945363 0.06771564 0.9968301 0.05104785 0.008080197 0.0007051509
## 310 0.09 0.005432685 0.06840034 0.9967668 0.05166704 0.008151336 0.0007120400
## 311 0.09 0.005968028 0.06893095 0.9967170 0.05213453 0.008229353 0.0007193280
## 312 0.09 0.006556124 0.06946373 0.9966667 0.05259205 0.008276857 0.0007239022
## 313 0.09 0.007202172 0.06997689 0.9966182 0.05302494 0.008343250 0.0007287593
## 314 0.09 0.007911882 0.07047994 0.9965696 0.05344150 0.008427799 0.0007357939
## 315 0.09 0.008691528 0.07098573 0.9965200 0.05383792 0.008507092 0.0007429860
## 316 0.09 0.009548001 0.07149538 0.9964697 0.05422345 0.008598959 0.0007511442
## 317 0.09 0.010488872 0.07204353 0.9964155 0.05462864 0.008681927 0.0007590260
## 318 0.09 0.011522457 0.07261606 0.9963588 0.05505809 0.008748189 0.0007655494
## 319 0.09 0.012657894 0.07319392 0.9963011 0.05549559 0.008830261 0.0007736313
## 320 0.09 0.013905217 0.07378927 0.9962412 0.05593664 0.008910537 0.0007818160
## 321 0.09 0.015275453 0.07439274 0.9961806 0.05636674 0.008983524 0.0007895651
## 322 0.09 0.016780713 0.07498028 0.9961220 0.05677520 0.009059795 0.0007975720
## 323 0.09 0.018434304 0.07558461 0.9960618 0.05718110 0.009147477 0.0008061893
## 324 0.09 0.020250841 0.07624365 0.9959962 0.05760335 0.009246672 0.0008150904
## 325 0.09 0.022246382 0.07700945 0.9959191 0.05809050 0.009352480 0.0008243512
## 326 0.09 0.024438566 0.07794587 0.9958228 0.05868649 0.009434784 0.0008319954
## 327 0.09 0.026846770 0.07887639 0.9957266 0.05922913 0.009518783 0.0008414413
## 328 0.09 0.029492282 0.07967560 0.9956455 0.05963326 0.009632989 0.0008517086
## 329 0.09 0.032398485 0.08046040 0.9955670 0.06000470 0.009758007 0.0008605140
## 330 0.09 0.035591069 0.08129369 0.9954837 0.06039580 0.009915162 0.0008725711
## 331 0.09 0.039098253 0.08219812 0.9953935 0.06080047 0.010082662 0.0008854428
## 332 0.09 0.042951040 0.08318572 0.9952951 0.06123359 0.010241693 0.0008970376
## 333 0.09 0.047183484 0.08422230 0.9951924 0.06169823 0.010422126 0.0009110963
## 334 0.09 0.051832999 0.08532880 0.9950830 0.06220846 0.010633694 0.0009266155
## 335 0.09 0.056940682 0.08653543 0.9949634 0.06283832 0.010884056 0.0009459823
## 336 0.09 0.062551681 0.08789117 0.9948280 0.06357285 0.011163940 0.0009697664
## 337 0.09 0.068715596 0.08944376 0.9946699 0.06443223 0.011463905 0.0009983773
## 338 0.09 0.075486909 0.09119400 0.9944888 0.06540613 0.011781453 0.0010311566
## 339 0.09 0.082925475 0.09316648 0.9942813 0.06654565 0.012144104 0.0010714880
## 340 0.09 0.091097046 0.09533704 0.9940508 0.06783612 0.012513782 0.0011145132
## 341 0.09 0.100073853 0.09768910 0.9937997 0.06926453 0.012916491 0.0011644322
## 342 0.09 0.109935245 0.10020104 0.9935306 0.07086426 0.013331664 0.0012168481
## 343 0.09 0.120768389 0.10287267 0.9932448 0.07263564 0.013764766 0.0012745794
## 344 0.09 0.132669044 0.10576896 0.9929333 0.07464046 0.014232043 0.0013424909
## 345 0.09 0.145742403 0.10886791 0.9925993 0.07680946 0.014696916 0.0014148688
## 346 0.09 0.160104026 0.11213695 0.9922486 0.07905130 0.015144322 0.0014911012
## 347 0.09 0.175880860 0.11551114 0.9918936 0.08133330 0.015477327 0.0015540461
## 348 0.09 0.193212361 0.11892057 0.9915550 0.08371203 0.015744789 0.0016111101
## 349 0.09 0.212251727 0.12237427 0.9912345 0.08616955 0.016060028 0.0016674040
## 350 0.09 0.233167255 0.12605835 0.9909060 0.08885904 0.016384702 0.0017229815
## 351 0.10 0.002331673 0.06503981 0.9970679 0.04865440 0.007756012 0.0006672514
## 352 0.10 0.002561438 0.06524769 0.9970496 0.04883565 0.007747857 0.0006695842
## 353 0.10 0.002813845 0.06547366 0.9970294 0.04903602 0.007759370 0.0006739406
## 354 0.10 0.003091125 0.06574939 0.9970050 0.04928570 0.007771806 0.0006782240
## 355 0.10 0.003395728 0.06609144 0.9969747 0.04959712 0.007822428 0.0006843920
## 356 0.10 0.003730347 0.06650349 0.9969380 0.04996992 0.007906292 0.0006920533
## 357 0.10 0.004097939 0.06699175 0.9968943 0.05041511 0.008001629 0.0007004432
## 358 0.10 0.004501755 0.06761647 0.9968381 0.05098900 0.008089839 0.0007080987
## 359 0.10 0.004945363 0.06820530 0.9967840 0.05152636 0.008186193 0.0007161074
## 360 0.10 0.005432685 0.06873230 0.9967348 0.05199447 0.008254574 0.0007224822
## 361 0.10 0.005968028 0.06928405 0.9966831 0.05247647 0.008296608 0.0007267970
## 362 0.10 0.006556124 0.06977639 0.9966362 0.05289698 0.008381846 0.0007339322
## 363 0.10 0.007202172 0.07026389 0.9965891 0.05330272 0.008462654 0.0007407980
## 364 0.10 0.007911882 0.07077470 0.9965393 0.05370269 0.008538173 0.0007471112
## 365 0.10 0.008691528 0.07129029 0.9964886 0.05408441 0.008623062 0.0007546825
## 366 0.10 0.009548001 0.07182245 0.9964364 0.05447459 0.008690736 0.0007609289
## 367 0.10 0.010488872 0.07237986 0.9963815 0.05488364 0.008751294 0.0007663946
## 368 0.10 0.011522457 0.07295106 0.9963246 0.05530719 0.008846978 0.0007751474
## 369 0.10 0.012657894 0.07354229 0.9962654 0.05574241 0.008925256 0.0007828314
## 370 0.10 0.013905217 0.07416631 0.9962026 0.05619523 0.008994579 0.0007910419
## 371 0.10 0.015275453 0.07477966 0.9961413 0.05662372 0.009069875 0.0007994425
## 372 0.10 0.016780713 0.07540290 0.9960794 0.05703814 0.009134662 0.0008056544
## 373 0.10 0.018434304 0.07605873 0.9960140 0.05746715 0.009212637 0.0008128068
## 374 0.10 0.020250841 0.07678546 0.9959411 0.05793566 0.009287513 0.0008190468
## 375 0.10 0.022246382 0.07762355 0.9958557 0.05846612 0.009371357 0.0008256062
## 376 0.10 0.024438566 0.07848122 0.9957670 0.05898964 0.009456095 0.0008341052
## 377 0.10 0.026846770 0.07926386 0.9956874 0.05940741 0.009566884 0.0008446035
## 378 0.10 0.029492282 0.08001696 0.9956119 0.05976847 0.009703059 0.0008552940
## 379 0.10 0.032398485 0.08081854 0.9955320 0.06014681 0.009832328 0.0008641931
## 380 0.10 0.035591069 0.08171702 0.9954419 0.06056320 0.009986313 0.0008761189
## 381 0.10 0.039098253 0.08267462 0.9953465 0.06098410 0.010135911 0.0008866180
## 382 0.10 0.042951040 0.08367018 0.9952472 0.06141811 0.010307593 0.0008993057
## 383 0.10 0.047183484 0.08471245 0.9951441 0.06188965 0.010521699 0.0009145196
## 384 0.10 0.051832999 0.08585847 0.9950302 0.06244517 0.010767122 0.0009330131
## 385 0.10 0.056940682 0.08713496 0.9949026 0.06313441 0.011035481 0.0009548786
## 386 0.10 0.062551681 0.08857866 0.9947565 0.06392688 0.011332501 0.0009820033
## 387 0.10 0.068715596 0.09022328 0.9945877 0.06485192 0.011650893 0.0010122111
## 388 0.10 0.075486909 0.09207501 0.9943952 0.06588969 0.012005197 0.0010485653
## 389 0.10 0.082925475 0.09415511 0.9941755 0.06709475 0.012371965 0.0010891659
## 390 0.10 0.091097046 0.09643176 0.9939321 0.06845965 0.012783992 0.0011391886
## 391 0.10 0.100073853 0.09884153 0.9936753 0.06994466 0.013208549 0.0011916792
## 392 0.10 0.109935245 0.10137125 0.9934050 0.07158686 0.013637361 0.0012464227
## 393 0.10 0.120768389 0.10408596 0.9931152 0.07343918 0.014103642 0.0013106820
## 394 0.10 0.132669044 0.10696850 0.9928076 0.07545967 0.014533747 0.0013771322
## 395 0.10 0.145742403 0.10998404 0.9924927 0.07754147 0.014921862 0.0014408992
## 396 0.10 0.160104026 0.11305549 0.9921826 0.07957776 0.015290439 0.0015027857
## 397 0.10 0.175880860 0.11619236 0.9918834 0.08171315 0.015580004 0.0015560725
## 398 0.10 0.193212361 0.11941857 0.9915934 0.08397107 0.015899402 0.0016076564
## 399 0.10 0.212251727 0.12289838 0.9912893 0.08644771 0.016223454 0.0016609059
## 400 0.10 0.233167255 0.12669266 0.9909694 0.08922393 0.016547997 0.0017167639
## 401 0.11 0.002331673 0.06516620 0.9970601 0.04881280 0.008306366 0.0007013016
## 402 0.11 0.002561438 0.06532288 0.9970469 0.04895033 0.008255644 0.0006972261
## 403 0.11 0.002813845 0.06556478 0.9970263 0.04915471 0.008180650 0.0006920659
## 404 0.11 0.003091125 0.06590468 0.9969959 0.04944915 0.008148009 0.0006940055
## 405 0.11 0.003395728 0.06628979 0.9969605 0.04978977 0.008138925 0.0007001948
## 406 0.11 0.003730347 0.06672013 0.9969207 0.05017185 0.008110257 0.0007039408
## 407 0.11 0.004097939 0.06733848 0.9968645 0.05071754 0.008098311 0.0007053304
## 408 0.11 0.004501755 0.06798312 0.9968053 0.05129767 0.008124815 0.0007087196
## 409 0.11 0.004945363 0.06852457 0.9967547 0.05179516 0.008194474 0.0007164497
## 410 0.11 0.005432685 0.06905446 0.9967051 0.05226617 0.008270843 0.0007231166
## 411 0.11 0.005968028 0.06955288 0.9966579 0.05268879 0.008348804 0.0007300457
## 412 0.11 0.006556124 0.07002656 0.9966123 0.05309169 0.008419214 0.0007359995
## 413 0.11 0.007202172 0.07052419 0.9965639 0.05350132 0.008495278 0.0007420767
## 414 0.11 0.007911882 0.07105006 0.9965123 0.05389878 0.008575131 0.0007495537
## 415 0.11 0.008691528 0.07157899 0.9964604 0.05428566 0.008643783 0.0007554689
## 416 0.11 0.009548001 0.07212537 0.9964066 0.05467395 0.008700322 0.0007605969
## 417 0.11 0.010488872 0.07268681 0.9963509 0.05508328 0.008772934 0.0007673383
## 418 0.11 0.011522457 0.07327605 0.9962920 0.05551497 0.008857333 0.0007757087
## 419 0.11 0.012657894 0.07391812 0.9962274 0.05598645 0.008939114 0.0007849437
## 420 0.11 0.013905217 0.07453996 0.9961652 0.05643063 0.008999598 0.0007920209
## 421 0.11 0.015275453 0.07515794 0.9961038 0.05684328 0.009063295 0.0007981487
## 422 0.11 0.016780713 0.07580193 0.9960397 0.05725948 0.009127217 0.0008039049
## 423 0.11 0.018434304 0.07651250 0.9959682 0.05771611 0.009214459 0.0008107003
## 424 0.11 0.020250841 0.07727969 0.9958905 0.05821004 0.009284957 0.0008164873
## 425 0.11 0.022246382 0.07809345 0.9958071 0.05874195 0.009348323 0.0008231231
## 426 0.11 0.024438566 0.07882121 0.9957331 0.05915324 0.009453580 0.0008328971
## 427 0.11 0.026846770 0.07953314 0.9956612 0.05950076 0.009597692 0.0008445017
## 428 0.11 0.029492282 0.08031295 0.9955832 0.05986991 0.009753176 0.0008561724
## 429 0.11 0.032398485 0.08119716 0.9954946 0.06028878 0.009889577 0.0008656751
## 430 0.11 0.035591069 0.08213463 0.9954006 0.06072304 0.010039064 0.0008764323
## 431 0.11 0.039098253 0.08310164 0.9953038 0.06115719 0.010213305 0.0008891277
## 432 0.11 0.042951040 0.08410432 0.9952043 0.06159661 0.010410423 0.0009027625
## 433 0.11 0.047183484 0.08520211 0.9950946 0.06211397 0.010647860 0.0009201522
## 434 0.11 0.051832999 0.08641972 0.9949726 0.06272754 0.010902741 0.0009411422
## 435 0.11 0.056940682 0.08779802 0.9948335 0.06346731 0.011174140 0.0009632034
## 436 0.11 0.062551681 0.08934097 0.9946756 0.06434204 0.011499617 0.0009932241
## 437 0.11 0.068715596 0.09106788 0.9944977 0.06531850 0.011848973 0.0010276818
## 438 0.11 0.075486909 0.09302863 0.9942918 0.06643280 0.012216816 0.0010651584
## 439 0.11 0.082925475 0.09515060 0.9940673 0.06763720 0.012621861 0.0011107163
## 440 0.11 0.091097046 0.09742672 0.9938247 0.06900956 0.013068137 0.0011649654
## 441 0.11 0.100073853 0.09980608 0.9935709 0.07053985 0.013488343 0.0012184260
## 442 0.11 0.109935245 0.10229584 0.9933107 0.07220797 0.013901142 0.0012735596
## 443 0.11 0.120768389 0.10498273 0.9930309 0.07404798 0.014376332 0.0013423585
## 444 0.11 0.132669044 0.10782978 0.9927381 0.07600020 0.014751022 0.0014001826
## 445 0.11 0.145742403 0.11067766 0.9924566 0.07790230 0.015110342 0.0014563789
## 446 0.11 0.160104026 0.11360037 0.9921843 0.07986176 0.015413320 0.0015059727
## 447 0.11 0.175880860 0.11661112 0.9919201 0.08191891 0.015726247 0.0015533506
## 448 0.11 0.193212361 0.11988393 0.9916409 0.08421548 0.016058948 0.0016041527
## 449 0.11 0.212251727 0.12348457 0.9913423 0.08675920 0.016394496 0.0016576243
## 450 0.11 0.233167255 0.12742178 0.9910252 0.08963382 0.016717016 0.0017132638
## 451 0.12 0.002331673 0.06479223 0.9970910 0.04862001 0.008446027 0.0007124853
## 452 0.12 0.002561438 0.06509334 0.9970652 0.04885646 0.008372732 0.0007088632
## 453 0.12 0.002813845 0.06547524 0.9970320 0.04916363 0.008280633 0.0007057736
## 454 0.12 0.003091125 0.06593987 0.9969909 0.04954239 0.008211528 0.0007047992
## 455 0.12 0.003395728 0.06642913 0.9969478 0.04992908 0.008140355 0.0007014028
## 456 0.12 0.003730347 0.06698459 0.9968975 0.05039443 0.008088852 0.0007014168
## 457 0.12 0.004097939 0.06768893 0.9968324 0.05103262 0.008151196 0.0007099059
## 458 0.12 0.004501755 0.06834310 0.9967713 0.05162021 0.008248120 0.0007194319
## 459 0.12 0.004945363 0.06886102 0.9967231 0.05207007 0.008284846 0.0007226657
## 460 0.12 0.005432685 0.06933889 0.9966781 0.05247885 0.008337511 0.0007267780
## 461 0.12 0.005968028 0.06980403 0.9966335 0.05287235 0.008417258 0.0007334903
## 462 0.12 0.006556124 0.07028255 0.9965868 0.05326613 0.008501593 0.0007408942
## 463 0.12 0.007202172 0.07077851 0.9965385 0.05365048 0.008578979 0.0007479575
## 464 0.12 0.007911882 0.07129552 0.9964881 0.05402778 0.008642524 0.0007539556
## 465 0.12 0.008691528 0.07182116 0.9964364 0.05440741 0.008705673 0.0007598689
## 466 0.12 0.009548001 0.07236735 0.9963823 0.05480469 0.008779029 0.0007667374
## 467 0.12 0.010488872 0.07296251 0.9963230 0.05523576 0.008863961 0.0007751113
## 468 0.12 0.011522457 0.07361923 0.9962571 0.05571979 0.008936951 0.0007832647
## 469 0.12 0.012657894 0.07425756 0.9961931 0.05618583 0.008989759 0.0007895270
## 470 0.12 0.013905217 0.07486538 0.9961327 0.05660397 0.009047524 0.0007953646
## 471 0.12 0.015275453 0.07550703 0.9960688 0.05702850 0.009117910 0.0008017851
## 472 0.12 0.016780713 0.07620566 0.9959986 0.05749082 0.009183600 0.0008075854
## 473 0.12 0.018434304 0.07693841 0.9959243 0.05798177 0.009244903 0.0008135155
## 474 0.12 0.020250841 0.07771095 0.9958456 0.05850460 0.009316912 0.0008209033
## 475 0.12 0.022246382 0.07841716 0.9957742 0.05893778 0.009399911 0.0008285948
## 476 0.12 0.024438566 0.07907716 0.9957080 0.05927377 0.009513211 0.0008367500
## 477 0.12 0.026846770 0.07982885 0.9956326 0.05963365 0.009652903 0.0008469294
## 478 0.12 0.029492282 0.08066933 0.9955485 0.06002896 0.009782545 0.0008552501
## 479 0.12 0.032398485 0.08158127 0.9954569 0.06045730 0.009940490 0.0008661295
## 480 0.12 0.035591069 0.08251622 0.9953630 0.06088260 0.010118892 0.0008789684
## 481 0.12 0.039098253 0.08350587 0.9952641 0.06132070 0.010305768 0.0008917751
## 482 0.12 0.042951040 0.08458191 0.9951558 0.06182225 0.010536806 0.0009096706
## 483 0.12 0.047183484 0.08576629 0.9950360 0.06239459 0.010789274 0.0009295300
## 484 0.12 0.051832999 0.08707946 0.9949029 0.06306844 0.011051237 0.0009503857
## 485 0.12 0.056940682 0.08851951 0.9947564 0.06385951 0.011344541 0.0009749588
## 486 0.12 0.062551681 0.09013005 0.9945909 0.06477504 0.011684382 0.0010070711
## 487 0.12 0.068715596 0.09195833 0.9944006 0.06580143 0.012064799 0.0010448748
## 488 0.12 0.075486909 0.09390122 0.9941976 0.06686789 0.012470614 0.0010873660
## 489 0.12 0.082925475 0.09598942 0.9939771 0.06808842 0.012881682 0.0011355775
## 490 0.12 0.091097046 0.09819616 0.9937466 0.06947717 0.013280114 0.0011848596
## 491 0.12 0.100073853 0.10053681 0.9935042 0.07098914 0.013736060 0.0012423608
## 492 0.12 0.109935245 0.10303917 0.9932460 0.07266875 0.014212332 0.0013071279
## 493 0.12 0.120768389 0.10575410 0.9929693 0.07452010 0.014568632 0.0013585570
## 494 0.12 0.132669044 0.10842215 0.9927093 0.07631379 0.014922301 0.0014122548
## 495 0.12 0.145742403 0.11116631 0.9924577 0.07814492 0.015215743 0.0014541658
## 496 0.12 0.160104026 0.11396442 0.9922158 0.08003361 0.015547309 0.0015019328
## 497 0.12 0.175880860 0.11703626 0.9919590 0.08212413 0.015878738 0.0015496538
## 498 0.12 0.193212361 0.12043172 0.9916820 0.08448505 0.016222032 0.0016009619
## 499 0.12 0.212251727 0.12415858 0.9913860 0.08712476 0.016547582 0.0016536854
## 500 0.12 0.233167255 0.12823752 0.9910719 0.09009004 0.016877171 0.0017108909
## MAESD
## 1 0.005409631
## 2 0.005409631
## 3 0.005409631
## 4 0.005409631
## 5 0.005409631
## 6 0.005409631
## 7 0.005377973
## 8 0.005328718
## 9 0.005270407
## 10 0.005207975
## 11 0.005150800
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## 13 0.005067736
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## 15 0.005099091
## 16 0.005157846
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## 19 0.005241352
## 20 0.005258061
## 21 0.005274202
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## 25 0.005432041
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## 27 0.005515124
## 28 0.005534394
## 29 0.005566259
## 30 0.005596479
## 31 0.005636104
## 32 0.005690036
## 33 0.005733849
## 34 0.005806730
## 35 0.005901141
## 36 0.006013286
## 37 0.006170839
## 38 0.006349378
## 39 0.006551668
## 40 0.006755814
## 41 0.007009178
## 42 0.007276577
## 43 0.007545443
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## 45 0.008111317
## 46 0.008438955
## 47 0.008786700
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## 93 0.007733424
## 94 0.008002690
## 95 0.008318328
## 96 0.008658152
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## 201 0.004993659
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## 399 0.010243928
## 400 0.010467814
## 401 0.005646656
## 402 0.005553625
## 403 0.005437730
## 404 0.005360514
## 405 0.005307996
## 406 0.005235328
## 407 0.005210034
## 408 0.005231219
## 409 0.005275222
## 410 0.005327076
## 411 0.005360524
## 412 0.005382878
## 413 0.005422309
## 414 0.005454768
## 415 0.005471555
## 416 0.005489502
## 417 0.005525277
## 418 0.005570949
## 419 0.005606238
## 420 0.005632598
## 421 0.005670504
## 422 0.005685715
## 423 0.005721930
## 424 0.005739544
## 425 0.005732268
## 426 0.005761066
## 427 0.005832549
## 428 0.005909481
## 429 0.005975883
## 430 0.006040471
## 431 0.006100840
## 432 0.006188958
## 433 0.006314353
## 434 0.006449528
## 435 0.006602115
## 436 0.006796917
## 437 0.007011935
## 438 0.007263008
## 439 0.007545796
## 440 0.007844062
## 441 0.008179116
## 442 0.008506911
## 443 0.008837320
## 444 0.009073942
## 445 0.009325063
## 446 0.009566158
## 447 0.009811639
## 448 0.010058993
## 449 0.010292833
## 450 0.010532931
## 451 0.005339752
## 452 0.005296911
## 453 0.005255910
## 454 0.005249416
## 455 0.005232885
## 456 0.005229190
## 457 0.005294380
## 458 0.005341979
## 459 0.005351427
## 460 0.005358708
## 461 0.005395147
## 462 0.005431369
## 463 0.005454653
## 464 0.005471385
## 465 0.005500509
## 466 0.005535455
## 467 0.005585091
## 468 0.005625554
## 469 0.005653086
## 470 0.005680168
## 471 0.005706056
## 472 0.005730967
## 473 0.005744854
## 474 0.005748746
## 475 0.005762252
## 476 0.005803307
## 477 0.005859475
## 478 0.005916020
## 479 0.005976505
## 480 0.006046429
## 481 0.006124487
## 482 0.006238495
## 483 0.006362767
## 484 0.006510739
## 485 0.006679567
## 486 0.006874877
## 487 0.007120624
## 488 0.007402774
## 489 0.007708539
## 490 0.008008162
## 491 0.008343102
## 492 0.008679472
## 493 0.008883685
## 494 0.009126227
## 495 0.009346557
## 496 0.009599318
## 497 0.009867230
## 498 0.010126813
## 499 0.010352763
## 500 0.010607286Neural network
set.seed(1234)
nnet_default <- caret::train( y ~ .,
data = dfii,
method = 'nnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl,
trace = FALSE)## Warning in nominalTrainWorkflow(x = x, y = y, wts = weights, info = trainInfo,
## : There were missing values in resampled performance measures.nnet_default$bestTune## size decay
## 9 5 0.1nnet_grid <- expand.grid( size = c(5,9,13,17,21),
decay = exp(seq(-6, 0, length.out = 11)))
set.seed(1234)
nnet_tune <- caret::train( y ~ .,
data = dfii,
method = 'nnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl,
tuneGrid = nnet_grid,
trace = FALSE)
nnet_tune$bestTune## size decay
## 47 21 0.008229747plot(nnet_tune, xTrans=log)
Random froest
registerDoParallel(cores=8)
set.seed(1234)
rf_default <- caret::train( y ~ .,
data = dfii,
method = "rf",
trControl = my_ctrl,
metric = my_metric,
importance = TRUE)
rf_default$bestTune## mtry
## 2 9rf_default$results## mtry RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 2 0.22429331 0.9767999 0.16492021 0.025547536 0.0044085567 0.017438581
## 2 9 0.06968765 0.9967131 0.05124449 0.007394860 0.0005828149 0.004599969
## 3 16 0.08476694 0.9949657 0.06163265 0.007351553 0.0008153420 0.005599433registerDoParallel(cores=8)
set.seed(1234)
rf_grid <- expand.grid(.mtry = (2:20))
rf_tune <- caret::train( y ~ .,
data = dfii,
method = "rf",
trControl = my_ctrl,
tuneGrid = rf_grid,
metric = my_metric,
importance = TRUE)
rf_tune$bestTune## mtry
## 7 8plot(rf_tune, xTrans=log)
Gradient boosted tree
registerDoParallel(cores=8)
set.seed(1234)
xgb_default <- caret::train(y ~ .,
data = dfii,
method = "xgbTree",
trControl = my_ctrl,
metric = my_metric,
verbosity = 0,
nthread = 1 )xgb_default$bestTune## nrounds max_depth eta gamma colsample_bytree min_child_weight subsample
## 33 150 2 0.3 0 0.8 1 0.75plot(xgb_default)
xgb_grid <- expand.grid(nrounds = seq(100, 2500, by = 300),
max_depth = c(3, 6, 9, 12),
eta = c(0.125, 0.25, 0.5) * xgb_default$bestTune$eta,
gamma = xgb_default$bestTune$gamma,
colsample_bytree = xgb_default$bestTune$colsample_bytree,
min_child_weight = xgb_default$bestTune$min_child_weight,
subsample = xgb_default$bestTune$subsample)registerDoParallel(cores=8)
set.seed(1234)
xgb_tune <- caret::train( y ~ .,
data = dfii,
method = "xgbTree",
trControl = my_ctrl,
metric = my_metric,
tuneGrid = xgb_grid,
verbosity = 0,
nthread = 1 )
xgb_tune$bestTune## nrounds max_depth eta gamma colsample_bytree min_child_weight subsample
## 8 2200 3 0.0375 0 0.8 1 0.75xgb_tune %>% plot()
#SVM
registerDoParallel(cores=8)
set.seed(1234)
svm <- caret::train( y ~ .,
data = dfii,
method = "svmRadial",
preProcess = c('center', 'scale'),
trControl = my_ctrl,
metric = my_metric)
svm$results## sigma C RMSE Rsquared MAE RMSESD RsquaredSD
## 1 0.04208154 0.25 0.1367684 0.9876522 0.10242319 0.014024510 0.002181577
## 2 0.04208154 0.50 0.1169683 0.9907386 0.08824700 0.010838741 0.001689743
## 3 0.04208154 1.00 0.1014462 0.9930260 0.07756597 0.009445028 0.001392402
## MAESD
## 1 0.009292271
## 2 0.007834135
## 3 0.007083717registerDoParallel(cores=8)
set.seed(1234)
sigma_values <- c(0.01, 0.03, 0.1, 1)
C_values <- c(0.25, 0.5, 1, 10, 100, 1000)
# Create the tuning grid
svm_grid <- expand.grid(sigma = sigma_values, C = C_values)
svm_tune <- caret::train( y ~ .,
data = dfii,
method = "svmRadial",
preProcess = c('center', 'scale'),
trControl = my_ctrl,
tuneGrid = svm_grid,
metric = my_metric)
svm_tune$bestTune## sigma C
## 5 0.01 100svm_tune$results## sigma C RMSE Rsquared MAE RMSESD RsquaredSD
## 1 0.01 0.25 0.12777795 0.9898228 0.09626653 0.015071645 0.0019356865
## 2 0.01 0.50 0.10896819 0.9920011 0.08193742 0.012590203 0.0015834416
## 3 0.01 1.00 0.09720543 0.9934760 0.07370577 0.010194209 0.0012359300
## 4 0.01 10.00 0.08017618 0.9957290 0.06513445 0.006808714 0.0008036555
## 5 0.01 100.00 0.07875694 0.9960889 0.06454914 0.006396148 0.0007903465
## 6 0.01 1000.00 0.07907856 0.9960585 0.06465774 0.006974880 0.0008243659
## 7 0.03 0.25 0.12936595 0.9889298 0.09713662 0.013829632 0.0020025915
## 8 0.03 0.50 0.11306541 0.9913085 0.08532935 0.010809715 0.0015570479
## 9 0.03 1.00 0.09883779 0.9933521 0.07585835 0.009012259 0.0012182063
## 10 0.03 10.00 0.08390817 0.9953818 0.06761251 0.008727717 0.0011020844
## 11 0.03 100.00 0.08335499 0.9954480 0.06720632 0.008360376 0.0010235638
## 12 0.03 1000.00 0.08361050 0.9954225 0.06729382 0.008013786 0.0009804075
## 13 0.10 0.25 0.17644351 0.9804149 0.12992477 0.018769715 0.0036338246
## 14 0.10 0.50 0.14522795 0.9859694 0.10905217 0.014388853 0.0029293035
## 15 0.10 1.00 0.12562099 0.9891770 0.09620281 0.013705532 0.0027085872
## 16 0.10 10.00 0.11202739 0.9912893 0.08594627 0.012756166 0.0022732438
## 17 0.10 100.00 0.11203188 0.9912965 0.08598945 0.012788991 0.0022808364
## 18 0.10 1000.00 0.11203188 0.9912965 0.08598945 0.012788991 0.0022808364
## 19 1.00 0.25 0.53583855 0.8806530 0.37533793 0.052490547 0.0268413572
## 20 1.00 0.50 0.37880698 0.9281586 0.25308602 0.046126011 0.0192836439
## 21 1.00 1.00 0.29744382 0.9480140 0.19900842 0.039281494 0.0158420011
## 22 1.00 10.00 0.28221300 0.9511905 0.18947958 0.037186028 0.0152540469
## 23 1.00 100.00 0.28221300 0.9511905 0.18947958 0.037186028 0.0152540469
## 24 1.00 1000.00 0.28221300 0.9511905 0.18947958 0.037186028 0.0152540469
## MAESD
## 1 0.010352226
## 2 0.009009789
## 3 0.007709334
## 4 0.005415330
## 5 0.005697106
## 6 0.006251034
## 7 0.009341009
## 8 0.008125951
## 9 0.007241491
## 10 0.006776076
## 11 0.006400757
## 12 0.006124480
## 13 0.011388911
## 14 0.008261096
## 15 0.007986662
## 16 0.007575849
## 17 0.007497325
## 18 0.007497325
## 19 0.032414152
## 20 0.025946183
## 21 0.020642183
## 22 0.020141208
## 23 0.020141208
## 24 0.020141208#PLS
registerDoParallel(cores=8)
set.seed(1234)
pls <- caret::train( y ~ .,
data = dfii,
method = "pls",
preProcess = c('center', 'scale'),
trControl = my_ctrl,
metric = my_metric)
pls$results## ncomp RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 1 0.2594890 0.9528249 0.1985486 0.02058900 0.007201300 0.01589521
## 2 2 0.1711272 0.9795025 0.1313449 0.01623436 0.003620325 0.01158483
## 3 3 0.1363831 0.9869397 0.1032318 0.01461153 0.002738435 0.01019301registerDoParallel(cores=8)
set.seed(1234)
ncomp_values <- 1:10
pls_grid <- expand.grid(ncomp = ncomp_values)
pls_tune <- caret::train( y ~ .,
data = dfii,
method = "pls",
preProcess = c('center', 'scale'),
trControl = my_ctrl,
tuneGrid = pls_grid,
metric = my_metric)
pls_tune$bestTune## ncomp
## 9 9pls_tune$results## ncomp RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 1 0.25948902 0.9528249 0.19854860 0.020589001 0.007201300 0.015895207
## 2 2 0.17112722 0.9795025 0.13134486 0.016234361 0.003620325 0.011584825
## 3 3 0.13638313 0.9869397 0.10323175 0.014611535 0.002738435 0.010193011
## 4 4 0.10206932 0.9927550 0.07769067 0.009472252 0.001344503 0.006968136
## 5 5 0.09567001 0.9936397 0.07096297 0.009358227 0.001257042 0.006499307
## 6 6 0.09316188 0.9939648 0.06942095 0.009363918 0.001195773 0.006103685
## 7 7 0.09229139 0.9940811 0.06838183 0.009404519 0.001201220 0.005987631
## 8 8 0.09100810 0.9942558 0.06736882 0.009045515 0.001147196 0.005989858
## 9 9 0.08916487 0.9944741 0.06714202 0.009046010 0.001097304 0.005961907
## 10 10 0.08918646 0.9944702 0.06718595 0.009105551 0.001106034 0.006023084caret_acc_compare <- resamples(list(lm_01 = train_lm_01,
lm_02 = train_lm_02,
lm_09 = train_lm_09,
lm_08 = train_lm_08,
enet_01 = enet_tune_01,
enet_09 = enet_tune_09,
enet_08 = enet_tune_08,
nnet_tune = nnet_tune,
rf_default = rf_default,
rf_tune = rf_tune,
xgb_default = xgb_default,
xgb_tune = xgb_tune,
svm_default = svm,
pls_default = pls,
pls_tune = pls_tune))dotplot(caret_acc_compare, metric = 'RMSE')
According to this question, the model “lm_08” has the lowest RMSE, the model with the lowest RMSE is generally considered to be the best, as it indicates the closest fit to the observed data, so here model “lm_08” is the